Melt butter in a nonstick skillet over medium heat. Cook and stir onion until starting to brown, about 5 minutes. Transfer to a plate.
Mix remaining 1 tablespoon butter, rosemary, onion powder, garlic powder, salt, and pepper in a small bowl. Spread on one side of each slice of bread.
Place 1 slice of bread buttered-side down in the skillet. Top with onion, mozzarella cheese, American cheese, and provolone cheese. Cover with second slice of bread, buttered-side up. Cook until golden brown, about 3 minutes. Flip with a spatula; cook until second side is browned and cheeses are melted, about 3 minutes more.