6 oz spaghetti
1/2 stick (1/4 cup) unsalted butter
8 large eggs
3 tablespoons olive oil
1 tablespoon whole milk
1/8 teaspoon freshly grated nutmeg
1/4 teaspoon salt
1/4 teaspoon black pepper
3 oz ham, cut into 1/4-inch dice
2 oz fresh mozzarella (not unsalted) or Fontina, cut into 1/4-inch dice
1/2 oz finely grated Parmigiano-Reggiano (1/4 cup) Accompaniment: tomato sauce

Cook spaghetti in a 6-quart pot of boiling salted water until al dente. Drain well in a colander, then return to pot and immediately toss with 2 tablespoons butter.

Lightly whisk together eggs, oil, milk, nutmeg, salt, pepper, ham, and mozzarella in a bowl.

Preheat broiler.

Heat remaining 2 tablespoons butter in a flameproof 12-inch nonstick skillet over moderate heat until foam subsides, then cook egg mixture, lifting up cooked edges with a heatproof spatula to let raw egg flow underneath, until set on bottom and almost set but still moist on top, 3 to 5 minutes. Arrange spaghetti over eggs, pressing lightly with spatula to help pasta settle into eggs, then sprinkle with Parmigiano-Reggiano.

Broil omelet 4 to 6 inches from heat until top is pale golden, 2 to 3 minutes.