Preheat oven to 400 degrees F (200 degrees C). Prick pie crust all over with a fork; place on a baking sheet.
Bake in the preheated oven until lightly golden, about 15 minutes. Remove crust and set aside to cool slightly; reduce oven temperature to 375 degrees (190 degrees C).
Bring a large pot of water to a boil and simmer leeks just until tender, about 7 minutes; drain thoroughly.
Whisk eggs, heavy cream, milk, chives, nutmeg, salt, and black pepper together in a bowl. Spread leeks evenly over bottom of pie crust; dot with goat cheese. Pour egg mixture over leeks.
Bake in the preheated oven until puffed, lightly browned, and a sharp knife inserted near the center comes out clean, 40 to 50 minutes. Remove to cool until warm or room temperature before serving.
