1/3 cup grated Parmesan cheese, plus a few Parmesan shavings for garnish
1/2 cup shredded basil
In a small, heavy saucepan, melt the butter in the milk and bring to a boil. Remove the pan from the heat. Add the flour all at once, mixing it in quickly with a sturdy wooden spoon. Return the pan to low heat and keep stirring; the mixture will form a solid mass. Continue cooking for about 1 minute to dry it somewhat.
Transfer the dough to the bowl of a food processor and process for 4 to 5 seconds. Add the eggs, salt, 3 tablespoons Parmesan cheese and nutmeg, then process for 15 to 20 seconds, until very smooth.
Heat about 3-inches of water in a large saucepan, but do not bring to a boil. If the dumplings boil, they will cook too fast, expand and eventually deflate. They should poach without expanding, since they will expand later when reheated on top of the stove or in the oven.
Using 1 1/2 to 2 teaspoons of dough per dumpling, scoop up the dough with a tablespoon, then push it with your index finger into the water in the saucepan; release the dough close to the water so it does not splash. Repeat, scooping and releasing the gnocchi as quickly as possible until all the dough is used up. Alternatively, fit a pastry bag with a 1-inch plain tip and fill the bag with the gnocchi dough. Rest the tip on the edge of the saucepan, press the dough out, and cut it off at the tip with a paring knife as the dough emerges, making small gnocchi, each approximately 1 1/2-inches long.
Poach the gnocchi in water for approximately 3 minutes. They will rise to the surface when they are cooked sufficiently. Using a slotted spoon, lift the gnocchi carefully from the water and place them in a bowl of ice water to cool. They will sink to the bottom of the bowl when cool. Drain and use right away, or refrigerate for later use.
At serving time, drop the gnocchi into a saucepan containing about 3 -inches of boiling water. Cook at a very gentle boil for 6 to 8 minutes, depending on size. Meanwhile, stir the melted butter and olive oil together in a small bowl. When the gnocchi are cooked inside and lightly puffed, transfer them with a slotted spoon to a warm platter. Toss lightly with the butter and oil, and sprinkle with the 1/3 cup grated cheese. Alternatively, preheat the oven to 375°. Arrange the cold gnocchi in one layer in one large or two smaller gratin dishes. Pour on the butter and oil, and top with the grated cheese. Bake for 20 to 25 minutes, until puffed and lightly browned. Sprinkle with the basil and cheese shavings, and serve.