Preheat oven to 450 degrees F (230 degrees C). Place pizza crust on a baking sheet.
Bring a small pot of lightly salted water to a boil. Cook cavatappi in the boiling water, stirring occasionally, until cooked through but firm to the bite, 10 to 11 minutes. Drain and return pasta to the pot.
Stir 1/2 the cheese sauce and butter into the cavatappi pot over medium heat until butter is melted and pasta-cheese mixture is combined. Season with salt and pepper.
Spread a thin layer of cheese sauce over the pizza crust. Sprinkle shredded Cheddar cheese over the cheese sauce onto the crust. Pour pasta-cheese mixture over the Cheddar cheese, and top with another 1/2 the cheese sauce.
Bake pizza in the preheated oven until golden and bubbling, 8 to 10 minutes. Let rest for 2 minutes before slicing.