I have read other posts (e.g. this) with descriptions of the thought process when playing long time controls, but I'm wondering what kind of information a GM processes during games with very short time controls.
Well, of course in very short time controls one cannot assess all the details prescribed in the answers of that good post. Grandmasters rely mostly on their vast experience, as much as it happens in other fields. But we can say that pattern recognition for tactics and mate threats play a major role. And they also learn to polish their intuition. They can efficiently and quickly prune the decision tree (even if they are wrong, they have the process automated) and they learn to recognize the critical points in which they should take some more time evaluating a position. They will have time saved for that critical moment, if it appears.
They look at lots of master games and repeat those same patterns, avoiding blunders.
In the opening, every move is used to get a piece out.
In the middle game, piece maneuvering occurs.
In the end game, they move their kings and rooks to the strongest positions.
There's no shortcut to getting good.