Whisk eggs, salt, and pepper together in a bowl until frothy. Melt 2 tablespoons butter in a large skillet over medium-low heat. Pour eggs into skillet; cook and stir with spatula until set, 3 to 6 minutes. Remove from heat.
Cut rolls in half horizontally; lay bottoms in a single layer in a 9x13-inch baking dish. Add a layer of provolone cheese, then scrambled eggs, then a layer of ham, then provolone cheese, then ham, then provolone cheese again. Cover with roll tops. Cut through ham and provolone cheese layers to make individual sandwiches.
Melt 1/4 cup butter, brown sugar, and mustard together in a small saucepan over medium heat; cook and stir until sugar is dissolved. Spread mixture over roll tops. Cover baking dish with aluminum foil and let stand, about 30 minutes.
Preheat oven to 350 degrees F (175 degrees C). Remove aluminum foil from baking dish.
Bake in preheated oven until tops are golden brown and cheese has melted, 15 to 20 minutes.
