I'm no expert but what you ask might not be feasible.
When I'm "thinking about how to solve a problem" -- I work as a software developer; and the conventional wisdom among programmers is that to be effective or "in the zone" you become immersed in the problem/subject/activity ... and the less you're distracted by anything else, e.g. by thoughts of other topics, or bodily sensations (including other people talking around you), or any "self-awareness" unrelated to the problem at hand, the better. The time for considering whether that activity is ethical might be before you embark on it -- because during it all you want to be aware of is the problem[s], the goal, and potential solutions.
Perhaps this is just me, or perhaps some similar state-of-mind might happen (and be sought as being desirable) during any slightly complicated or dangerous activity -- driving a car or riding a bicycle, performing surgery or carpentry, etc.
Such intense or focussed awareness isn't just a matter of thinking more about the subject, it's also a matter of think less about any distractions irrelevant to the problem (and it takes time to become immersed at the start of a session, for cessation of mental activity unrelated to the problem; that immersion includes putting aside some notions of self -- including "I don't like this problem" and "it's too difficult for me" -- and is I think part of the skill or habit which a student or practitioner or expert develops, that learned ability is as well as the necessary body of knowledge).
As such you have the problem you raised -- that being "mindful" would interrupt the activity.
I think that when I'm working I'm mindful of the current problem[s] and the goal -- and potential solutions (to the problems) come to mind -- and then as long as I'm aware of the viable solution, I keep that in mind as I act to implement it. The subject (details of what I'm thinking about) changes -- as new solutions come to mind; as I encounter problems with implementing a specific solution; as solutions are completed and I move to a next phase (e.g. testing or another problem) -- or if I lose my concentration, if somebody talks to me, or if I need to sleep, or whatever it is.
This mode of thinking is all very well, it's practical; and if you're careful it's ideally harmless too and beneficial in a mundane or "householder" way.
There's another but related mode of thinking, when the problem exists but when a solution isn't obvious. There's a description of that -- I like the description -- in Zen and the Art of Motorcycle Maintenance (which the Author's Note says, "should in no way be associated with that great body of factual information relating to orthodox Zen Buddhist practice"):
Well, those were the commonest setbacks I can think of: out-of-sequence reassembly, intermittent failure and parts problems. But
although setbacks are the commonest gumption traps they’re only the external cause of gumption loss. Time now to consider some of
the internal gumption traps that operate at the same time.
As the course description of gumptionology indicated, this internal part of the field can be broken down into three main types of
internal gumption traps: those that block affective understanding, called "value traps"; those that block cognitive understanding, called
"truth traps"; and those that block psychomotor behavior, called "muscle traps." The value traps are by far the largest and the most
dangerous group.
Of the value traps, the most widespread and pernicious is value rigidity. This is an inability to revalue what one sees because of
commitment to previous values. In motorcycle maintenance, you must rediscover what you do as you go. Rigid values make this
impossible.
The typical situation is that the motorcycle doesn’t work. The facts are there but you don’t see them. You’re looking right at them, but
they don’t yet have enough value. This is what Phædrus was talking about. Quality, value, creates the subjects and objects of the world.
The facts do not exist until value has created them. If your values are rigid you can’t really learn new facts.
This often shows up in premature diagnosis, when you’re sure you know what the trouble is, and then when it isn’t, you’re stuck. Then
you’ve got to find some new clues, but before you can find them you’ve got to clear your head of old opinions. If you’re plagued with
value rigidity you can fail to see the real answer even when it’s staring you right in the face because you can’t see the new answer’s
importance.
The birth of a new fact is always a wonderful thing to experience. It’s dualistically called a "discovery" because of the presumption that
it has an existence independent of anyone’s awareness of it. When it comes along, it always has, at first, a low value. Then, depending
on the value-looseness of the observer and the potential quality of the fact, its value increases, either slowly or rapidly, or the value
wanes and the fact disappears.
The overwhelming majority of facts, the sights and sounds that are around us every second and the relationships among them and
everything in our memory...these have no Quality, in fact have a negative quality. If they were all present at once our consciousness
would be so jammed with meaningless data we couldn’t think or act. So we preselect on the basis of Quality, or, to put it Phædrus’
way, the track of Quality preselects what data we’re going to be conscious of, and it makes this selection in such a way as to best
harmonize what we are with what we are becoming.
What you have to do, if you get caught in this gumption trap of value rigidity, is slow down...you’re going to have to slow down
anyway whether you want to or not...but slow down deliberately and go over ground that you’ve been over before to see if the things
you thought were important were really important and to—well—just stare at the machine. There’s nothing wrong with that. Just live
with it for a while. Watch it the way you watch a line when fishing and before long, as sure as you live, you’ll get a little nibble, a little
fact asking in a timid, humble way if you’re interested in it. That’s the way the world keeps on happening. Be interested in it.
At first try to understand this new fact not so much in terms of your big problem as for its own sake. That problem may not be as big as
you think it is. And that fact may not be as small as you think it is. It may not be the fact you want but at least you should be very sure
of that before you send the fact away. Often before you send it away you will discover it has friends who are right next to it and are
watching to see what your response is. Among the friends may be the exact fact you are looking for.
After a while you may find that the nibbles you get are more interesting than your original purpose of fixing the machine. When that
happens you’ve reached a kind of point of arrival. Then you’re no longer strictly a motorcycle mechanic, you’re also a motorcycle
scientist, and you’ve completely conquered the gumption trap of value rigidity.
I find it an amusing description of how solutions (or "facts") sometimes come to mind.
It can sometimes happen that you're "barking up the wrong tree", maybe too focussed on a unworkable solution, or maybe you've forgotten something important -- in that case, putting your tools down, and going for a walk, and not thinking about the problem for a while provides an opportunity -- a mind that's not too busy or too focussed in a wrong direction -- for a potentially helpful solution to "pop up" or arise.
Then there's "mindfulness" as it's taught in Buddhism. One of the classic summaries is here: Anussati — The Recollections. I'm not sure how meaningful they are, I suspect they act partly as a summary of everything you've learned about and learned from Buddhism -- which, I assume, varies from person to person. Still I find that it i.e. Buddhism is a good "solution" to a lot of the "problems" of life. Maybe it won't fix your "motorcycle", maybe it will help peace of mind as you work to repair it, or to "let go" of it, and to "value" it correctly.
Another thing is that a lot of what we work with, the world we live is, is a "fabrication" -- Can anyone explain Sanskara / Sankara indepth?
To some extent, that's apparently normal and there's no avoiding it, and so the question is about the quality of it (like whether it's ethical), and one's reaction (like whether one tries to attach to it) -- see for example the Dhammapada:
- All mental phenomena have mind as their forerunner; they have mind as their chief; they are mind-made. If one speaks or acts with an evil mind, 'dukkha' (suffering etc.) follows him just as the wheel follows the hoofprint of the ox that draws the cart.
- All mental phenomena have mind as their forerunner; they have mind as their chief; they are mind-made. If one speaks or acts with a pure mind, happiness (sukha) follows him like a shadow that never leaves him.
I think of the kind of "applied" thought, that I described at the start of this above, is like a kind of vehicle -- it's a means of getting from A to B, from a problem to a finished solution. But if you do have a vehicle like that, you might not want to be driving it all the time -- spending your whole life, pushing the engine on a highway -- even if only because it's not sustainable, you ought to slow down sometimes (when the going gets difficult), maybe cool down or empty your mind to make it receptive (like a muscle needs periods of food and rest in order to work) etc. I'm told it's also useful to understand something of a relationship between mind and body -- like what does "feeling anxious" mean, exactly, what is "peace", what is "impermanence"?
Anyway, for whatever reason, one of the types of meditation people teach or practice is "mindfulness" -- "mindfulness of breathing" for example, "mindfulness of the body", and so on.
I'm not sure those can be practiced at the same time as mundane problem solving. Maybe yes during "physical" activity, maybe less so during some sustained/applied mental activity -- or maybe other people can and it's just my inability. But I think there's a reason why people set aside time -- dedicate some time to "meditating" -- or go on a "retreat", for a week-end or ten days -- or become like monks, that might be so that they can concentrate on the practice literally full-time.
I'm not sure but maybe the practice is intended as an antidote to -- deliberately, as a way to interfere with -- the kind of immersion with mundane worries and concepts which may normally occupy a mind.
There's also a suggestion that, when you're not preoccupied with mental activity, then all that remains to be aware of is the physical -- so trying to be aware (or mindful) of the physical is a way to, as this answer puts it, to "work within the framework of the target state". And it's a respite from perpetually driving the mental engine.
Maybe it's also good to be mindful of the body "occasionally" during mundane activity. The breath becoming "short" for example might be a symptom of the mind becoming too tight -- and correcting that might improve the mind's interaction with the activity.
