Whisk the eggs, cream, milk, cayenne, nutmeg, paprika, salt, and pepper to taste in a bowl. Toss the 1 cup cheddar, 1/2 cup havarti dill, and 1/2 cup pecoreno romano in a separate bowl.

Bring a large pot of salted water to a boil. add the cauliflower and cook until almost falling apart, about 7 minutes. Transfer with a slotted spoon to a bowl. Add the pasta to the same water and cook until al dente, about 10 minutes. Drain, reserving 1/4 cup cooking water. Preheat broiler.

Combine the egg mixture and the grated cheeses in the empty pot and cook over medium-low heat, stirring constantly, until the cheese melt and the sauce begins to thicken. Remove from the heat and add the cauliflower. Puree with an immersion blender until smooth and light (you could also use a regular blender). Stir in some of the reserved pasta water until creamy.

Toss the pasta in the sauce; season with salt and black pepper. Transfer to a shallow casserole dish and top with 1/3 cheddar and 1/4 pecorino romano. Broil until golden brown, about 5 minutes.