Bring a pot of salted water to a boil, add spaghetti and cook according to package directions, stirring often, until al dente. Drain pasta and rinse well with cold water.
Preheat oven to 400°F. Grease an ovenproof 9- or 10-inch skillet with about 1/2 Tbsp. butter.
In a bowl, whisk together eggs, milk, bacon, scallion, pepper and 3 Tbsp. Parmesan. Add spaghetti; mix well. Transfer mixture to skillet, spreading evenly. Dot with remaining butter and sprinkle remaining 1 Tbsp. Parmesan on top. Bake for 25 to 30 minutes, until center is set and top is golden. Let cool for 10 minutes before serving.