The question in the title is far from trivial.
I'm doing quite a lot of blitz problems at CT, lately, in an attempt to implement my new thought process. A thought process is maintained by thoughts, and thoughts are notoriously slow by nature.
Sometimes I recognize something in the position, and I let trial and error take over. If I'm lucky, I find the solution, but more often than not, I find that I continue to use up time, without remembering that I was busy with a thought process. The conclusion must be: during this stage of implementation of the thought process, I must suppress the inclination to use T&E.
The thoughts itself can't be sped up. I asked myself if there could be any salt mine exercise useful to speed up the thought process. I estimated that this might be useful in less than 2% of the cases.
The only way to speed up the thought process is to build new cues and patterns by thoroughly analysing the positions, and categorize them in positions with equal characteristics. I can't see no other way.