In a small bowl, whisk together the eggs, salt, and pepper.
Melt 1 tablespoon of the butter in a large nonstick skillet over medium heat. Pour in the eggs and scramble until cooked through.
Divide the eggs among the bottom halves of the rolls. Add the prosciutto and cheese and sandwich with the tops of the rolls.
Melt the remaining butter in a grill pan or large nonstick skillet over medium heat. Add half the sandwiches to the skillet. Cook, pressing frequently with the back of a spatula or placing another pan on top of the sandwiches to weigh them down, until the cheese has melted and the bread is golden, 4 to 6 minutes. Repeat with the remaining sandwiches.