Bake-and-Slice Macaroni and Cheese

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1 tablespoon unsalted butter, plus more for the dish

Kosher salt

8 ounces elbow macaroni

2 cups half-and-half

3 large eggs

1/2 teaspoon paprika

Freshly ground pepper

2 cups grated cheddar cheese (about 6 ounces)

1 cup grated muenster cheese (about 3 ounces)

1 cup grated provolone cheese (about 3 ounces)

1/2 cup finely chopped whole-milk mozzarella cheese (about 2 ounces)