Preheat the oven to 400 degrees F. Coat a 13 by 9 by 2-inch baking dish with 1 tablespoon of butter and set aside.
Cook the potatoes in a large pot of boiling salted water until they are very tender, about 15 minutes. Drain; return the potatoes to the same pot and mash well. Mix in the milk and melted butter. Mix in the mozzarella and 3/4 cup of the Parmesan. Season, to taste, with salt and pepper. Transfer the potatoes to the prepared baking dish. Stir the bread crumbs and remaining 1/4 cup of Parmesan in a small bowl to blend.
Sprinkle the bread crumb mixture over the mashed potatoes. Recipe can be prepared up to this point 6 hours ahead of time; cover and chill.
Bake, uncovered, until the topping is golden brown, about 20 minutes.