Heat a nonstick skillet over medium heat with the butter. When the butter is melted. Lightly moisten both pieces of bread in the butter. Top one slice of bread with the provolone and tomato and cover with the other slice of bread. Cook until golden brown on both sides, flipping a couple of times.
When the basic grilled cheese is done, remove it from the pan. Turn the heat to medium. If the pan seems very dry, melt a touch more butter in it. Sprinkle the Parmigiano in the still-hot skillet, in a shape roughly the same as the sandwich.
Now watch closely, as the cheese melts. You'll see the fat start to cook out and the cheese begin to brown and crisp. When it has just begun to brown, put the sandwich back on top and press down with a large spatula to weld the frico to the bread. Let cook for one more minute then serve.