Two minutes a side is far too long, and the fix is to make the sear violently short rather than to make the pan cooler.
What works for me on a 1 inch steak:
- rest the bagged steak, then take it out, pat it dry and leave it uncovered on a rack in the fridge or freezer for 10 to 15 minutes. A cold, dry surface buys you time before the interior notices
- get the pan to the point where a drop of high smoke point oil is shimmering and just starting to smoke
- 45 to 60 seconds total per side, flipping every 20 seconds rather than leaving it still
- finish with butter off the heat, not during
That gives a proper crust with a grey band you have to look for. Flipping often sounds fussy but it genuinely keeps the sub surface cooler while the surface still browns.