He's not doing this cold, remember. First of all he's coming out of a tournament game, so he's already in chess mode. That helps a lot.
Secondly, they're following everyone else's games as they play. Not in depth, but they get up, walk around, and check what openings everyone has played, then what strategic/tactical things are happening. You hear them all the time say something like "30 minutes ago I saw he was in trouble, I'm surprised it's so equal now"
He's also one of the strongest GMs in the world. Study chess for 8 hours a day for 10 years and you're not going to have to count individual pawns, you'll see whole structures all at once and just from this you'll understand dozens of strategic and tactical implications all at once.
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Anyway, to answer your question, it's a chicken and egg question. Do you calculate first or do some static evaluations? It's always back and forth. You do a little of each over and over, each time a little longer than the last. And it has to be said there are SO MANY unconscious patters that help you evaluate and calculate. There are so many and it's so intuitive it can't even be expressed if someone tried.
By the way all this may happen in just a few seconds depending on the player and the position.
Anyway, after the initial short calculations and static evals (and patter recognition) you'll have a feel for the position and only then (and if the position requires it) you do any long calculation or careful positional analysis.
This was some months ago and I was watching an interview of Fabio Caruana on Chess.com/tv after one of his games. I think it was the Isle of Man Tournament. Anyways, they were showing Fabio some positions from games on other boards, and he would like instanteously start rattling off lines from his supersonic calculating speed about what's a good move in a position.
And I'm like, "Holy Moly, how does he do that? I'm still looking at the position and just trying to get the material count, and what's hanging, and what's strong and what's weak, and I'm just basically still counting the number of pawns on each side while he's already rattling off lines and variations!"
I'm like, how does he do that?!!! Aren't you supposed to, like, you know, look at the position first, AND THEN, start looking at moves? Have I been doing it wrong and in the wrong order? Should I just start calculating concretely, and then hope that the positional and strategic essence will come to me at some time later during my calculations?
What do you guys do? Someone shows you a position and you just start calculating right away? Or do you get the lay of the land first, and then start looking at lines afterwards?