Heat a nonstick skillet over medium heat and add the butter, melting it until foamy.
Lightly coat both sides of each bread slice with the melted butter using the pan or a brush.
Place provolone slices on one buttered slice of bread, top with tomato slices, then cover with the second slice of bread, buttered side out.
Cook the sandwich in the skillet over medium heat until the bottom is golden brown, then flip and cook the other side until golden, flipping a couple of times as needed to melt the cheese, about 3–4 minutes per side.
Remove the sandwich from the pan and reduce heat to medium. If the skillet is very dry, add a little more butter and let it melt.
Evenly sprinkle the grated Parmigiano-Reggiano in the hot skillet in a circle roughly the size of the sandwich and let it melt and begin to brown and crisp around the edges.
When the frico has just begun to brown, place the sandwich back on top of the melted cheese and press down with a spatula to adhere the crisped cheese to the bread; cook about 1 more minute.
Remove from the skillet, let sit briefly to set, then slice and serve.