Preheat oven to 425°F.
Heat a large, nonstick skillet with an oven-safe handle over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Add the leftover pasta and cook until lightly browned, 2-3 minutes.
While the pasta is browning, beat the eggs with the milk, Parmigiano Reggiano and some salt and pepper.
Add the egg mixture to the skillet and stir gently to combine. Allow the eggs to set up on the bottom of the pan and become firm. Transfer the skillet to the oven and bake until the eggs have set and the top is golden brown, about 10 minutes.