You're misunderstanding what's being described - there's nothing monk like about it.
Using myself as an example, you have a couple techniques you're good at (linear programming) and a couple of problems you've tried to solve before but failed (fixed parameter tractable problems). Every time someone tells you about a problem (or you run into one) you try apply the thing you're good at ("will an LP work here"). Every time someone tells you about an approach they took with their problem you try it out on your thing.
It all happens completely automatically/naturally/fluently if you're good at the technique and have actually tried to solve said pet problem. Like I don't need to write code or calculate anything because I know by now exactly where/what my blockers. Conversely I know very quickly when an LP is appropriate because I know the assumptions/requirements very well.
Keep a few significant problems in your mind...and also keep a few significant solutions / problem solving techniques in your mind. Then when you encounter new problems, check them against your set of solutions and see if any of them apply. Also, when you encounter new problem solving techniques, check them against your set of problems to see if they're applicable. Whenever you encounter a new problem or solution that seems unusually significant, add it to the list that you keep track of
This is verbatim how Feynman describes it in "surely you're joking" (the chapter with spinning plate and QED iirc).
"Most great scientists know many important problems. They have something between 10 and 20 important problems for which they are looking for an attack. And when they see a new idea come up, one hears them say ``Well that bears on this problem.'' They drop all the other things and get after it. "
Solve these:
or a question to kick start this like:
What is the solution to these problems:
You'll quickly reach the point where you are ignoring what is written, but the subtle visual reminder will keep a small portion of your mind churning away at them.