Heat oven to broil and arrange rack in top. Butter a 9 by 9-inch baking dish and set aside. Melt 1 tablespoon of the remaining butter and mix in a medium bowl with 1 cup of the Parmigiano and all the bread crumbs until thoroughly moistened; set aside.

Bring a large pot of heavily salted water to a boil. Add pasta and cook for half the time indicated on package.

Meanwhile, melt remaining 2 tablespoons butter in a large frying pan over medium heat. When it starts to foam, add sage and cook until crisp and butter begins to turn golden brown. Remove from heat and set aside until pasta is ready. Reserve 1 cup pasta water and drain pasta.

Return pasta to pot and place over low heat. Stir in sage butter, reserved pasta water, remaining 3 cups Parmigiano, Gruyere, Cheddar, mascarpone, and salt, and season with freshly ground black pepper. Stir constantly until cheeses are evenly melted and the pasta looks well coated. Turn pasta into baking dish and evenly top with bread crumb mixture. Place under broiler until mixture bubbles and top is browned, about 1 to 2 minutes. Let sit 5 minutes before serving.