In a small bowl, whisk the milk with the egg yolks and egg. In a large bowl, whisk the flour with the nutmeg, 1 teaspoon of salt and 1/4 teaspoon of pepper. Using a wooden spoon, stir the egg mixture into the flour, leaving a few lumps. Cover and refrigerate the batter for at least 1 hour or overnight.

Bring a large pot of salted water to a boil. Prepare a large bowl of ice water. Carefully hold a colander with large holes over the boiling water. Add about 1/2 cup of the batter to the colander and press it into the simmering water with a spatula or the back of a spoon. Repeat until all of the batter has been used. Cook the spaetzle for 2 minutes longer, then drain. Immediately transfer the spaetzle to the ice water, swirling the dumplings until all of the ice melts. Drain and transfer to a large bowl. Stir in 1 tablespoon of the oil, season with salt and pepper and toss to coat.

Preheat the oven to 400°. Oil a 9-by-13-inch baking dish. Spread the spaetzle in the dish and dot with the butter. Sprinkle with the cheese and bake for about 20 minutes, or until the spaetzle is hot and the cheese is just melted.