Melt 1 tablespoon butter in a large cast-iron skillet over medium heat, add the onion and a pinch of salt, and cook for 12-15 minutes until soft and deeply golden.

Increase heat to medium-high, add the remaining butter, and sear the shaved steak for 1 minute without moving, then toss for another minute until no longer pink. Season with salt and pepper.

Lower heat to low, divide the steak into two piles, top each with 2 slices provolone, cover the pan, and melt the cheese for 30 seconds to 1 minute.

Toast the split rolls cut-side down in a dry pan until golden, then pile on the cheesy steak and onions. Slice each sandwich in half and serve right away.