Cook cavatappi in a large pot of water with the tablespoon salt until just tender. Reserve 1/2 cup pasta water and drain.
While the pasta cooks, melt butter in a saucepan over medium heat and whisk in flour for 1 minute. Gradually add milk, then sauce salt and pepper.
Simmer for 4 to 5 minutes until the sauce coats a spoon, then remove from the heat.
Stir in fontina and cheddar by handfuls, followed by mozzarella and finally Parmesan, letting each addition melt before the next.
Fold in cavatappi and loosen with reserved pasta water 1 tablespoon at a time if needed. Serve while the sauce is smooth and gently stretchy.