In a small bowl, whisk together the eggs, ½ teaspoon salt, and ¼ teaspoon 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. Top with the prosciutto and cheese and sandwich with the tops of the rolls.
Melt the remaining tablespoon of butter in a grill pan or large nonstick skillet over medium heat. Place two sandwiches in the pan.
Cook, pressing often with a spatula or placing a pan on top to weigh down the sandwiches, until cheese has melted and bread is golden, 4 to 6 minutes.
Repeat with the remaining sandwiches.