Preheat the oven to 425°F (220°C).

Bring a pot of salted water to a boil.

Cook the spaghetti in the boiling water until al dente.

Drain the spaghetti thoroughly.

In a large bowl, combine the cooked spaghetti, milk, grated Parmigiano-Reggiano cheese, beaten eggs, ground black pepper, and kosher salt.

Add 1 1/2 cups of shredded Fontina cheese and 1 1/2 cups of shredded sharp white cheddar cheese to the mixture.

Mix all ingredients until well combined.

Grease a 9-inch springform pan with butter.

Scrape the pasta mixture into the prepared springform pan.

Sprinkle the remaining 1/2 cup of Fontina cheese and 1/2 cup of cheddar cheese evenly over the top of the pasta pie.

Bake in the preheated oven for 35 to 40 minutes, or until the cheese is melted, bubbling, and the pie is set.

Turn on the broiler.

Broil the pie 8 inches from the heat for 2 to 3 minutes, or until the top is browned to your liking.

Transfer the pasta pie to a wire rack and let it cool for 15 minutes before serving.

Remove the ring from the springform pan.

Cut the pie into wedges.

Serve immediately.