Preheat the oven to 350 degrees F. Butter a shallow 2-quart baking dish; set aside. Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs for al dente. Drain and transfer to a medium bowl; add the butter and toss until melted.
Whisk the half-and-half, eggs, paprika, 1 teaspoon salt and a few grinds of pepper in a large bowl until smooth. Add the pasta, 1 1/2 cups cheddar, the muenster, provolone and mozzarella; toss.
Transfer the pasta mixture to the prepared baking dish and sprinkle with the remaining 1/2 cup cheddar. Bake until bubbling around the edges and no longer jiggly, 30 to 35 minutes. Let cool about 30 minutes before slicing.