1. Bring a large pot of water to a boil, then add pasta; salt generously and cook pasta to manufacturer's instructions. Transfer to a colander, rinse under cold running water, and drain well.

2. Melt butter in a 4-quart pot over medium heat. Whisk in flour and cook, stirring with a wooden spoon, until bubbling but not browning, about 45 seconds. Add milk and whisk to combine. Bring to a simmer, stirring with a wooden spoon (scrape across the bottom and around edge of pot to prevent scorching), until thickened and smooth. Reduce heat to a simmer and cook for 20 minutes.

3. Add fontina, Gruyere, Cheddar, and Parmigiano-Reggiano cheeses, stirring until completely melted and sauce is smooth. Season with salt and black pepper, add cayenne and nutmeg, and stir to combine.

4. Add pasta to sauce along with white pepper and salt; stir to thoroughly combine. Serve immediately.