Bring milk and ¼ cup melted butter to a boil in a medium saucepan over medium-high. Remove from heat, and add flour all at once, stirring quickly using a sturdy wooden spoon. Return pan to heat over low, stirring constantly until mixture forms a solid mass. Cook, stirring constantly, until mixture is somewhat dry, about 1 minute.

Transfer dough to bowl of a food processor. Process until slightly cool, about 10 seconds. Add eggs, 3 tablespoons Parmesan, salt, and nutmeg; process until dough is very smooth and paste-like, 15 to 20 seconds.

Fill a large pot with salted water to a depth of 3 inches. Bring to a boil over high; reduce heat to medium-low to maintain a very gentle simmer.

Spoon dough into a piping bag fitted with an 11/16-inch-round piping tip. Working in 3 to 4 batches, rest the piping tip on edge of pot to position it over the simmering water and press dough out, slicing off 1- to 1 ½-inch-long pieces using a paring knife. (Alternatively, scoop 2 teaspoons dough using a large spoon, and push dough off spoon into water using your finger.) Make sure to release dough close to the water surface to avoid splashing.

Poach gnocchi in simmering water until they rise to the surface, 2 to 3 minutes. (Avoid allowing the water to reach a boil; the gnocchi will cook too fast, expand, and eventually deflate. Gnocchi should poach without expanding, since they will expand later when reheated in the oven.) Meanwhile, prepare a bowl of ice and water. Carefully lift gnocchi from water using a slotted spoon; place in ice bath, and let cool. (Gnocchi will sink to bottom of bowl when cool.) Drain and pat gnocchi dry using paper towels; use immediately, or refrigerate.

Preheat oven to 375°F. Arrange cooled gnocchi in a single layer in an 11- x 8-inch, 6½-cup gratin dish. Drizzle with oil and remaining 2 tablespoons melted butter; sprinkle evenly with remaining ⅓ cup Parmesan.

Bake in preheated oven until puffed and golden brown, 25 to 30 minutes. (Gnocchi will deflate slightly out of oven.) Garnish with basil and shaved Parmesan. Serve immediately.