Preheat the oven to 425 degrees F.

Grease a small baking sheet with 1 teaspoon of the butter and set aside. Butter a piece of parchment paper the same size as the baking sheet with 1/2 teaspoon of the butter and set aside, buttered side-up. Butter a 9-inch square baking pan with 1 teaspoon of butter and set aside.

Combine the milk, salt, pepper, and nutmeg in a medium, heavy pot. Bring to a boil. Add 1/4 cup of the semolina, and whisk constantly until the mixture begins to thicken. Lower the heat to medium-low. Continue to cook, stirring constantly with a heavy, wooden spoon, adding the remaining semolina flour 1/4 cup at a time, about 12 minutes. The mixture should be very stiff and thick. Remove from the heat and let rest 5 minutes.

Beat the eggs in a large bowl. Add 1/2 cup of the semolina mixture to the eggs and whisk well to incorporate. Add the eggs to the remaining semolina mixture and stir as hard as possible to blend. Add 1/4 cup of the cheese and stir well to blend. Turn out the mixture onto the prepared baking sheet and using an off-set cake spreader or rubber spatula, spread across the bottom. Place the buttered parchment paper directly on the batter. With an outward motion, spread the batter evenly across the pan and smooth the top. Refrigerate until completely cooled, 20 to 25 minutes.

With a knife, cut the batter into 24 equal pieces, about 1 1/2-inches square, dipping the knife in hot water as needed to cut cleanly. Lay the pieces in 4 rows in the prepared 9-inch pan, shingling them to fit. Dot the top with the remaining butter and sprinkle with the remaining 1/2 cup of cheese. Bake until the top begins to brown and the gnocchi are puffed, about 25 minutes.

Remove from the oven and let stand for 5 minutes before serving.