Gougeres take their flavor from Gruyere cheese.
Preheat oven to 425 degrees with a rack in the center. Line two unrimmed baking sheets with parchment paper or Silpats (nonstick French baking mats).
Make Pate a Choux batter; add cheese, salt, pepper, and nutmeg at the end. (If too stiff, add teaspoonfuls of water until the proper consistency is reached.) Fill a pastry bag fitted with filling tip (Ateco #804); pipe 3/4-inch rounds that are about 1/2 inch high (about the size of gumdrops) onto baking sheets at 2-inch intervals. Use your finger to gently coat entire top with egg wash, being careful not to let it drip onto the surrounding baking sheet (it will inhibit rising). Sprinkle with grated cheese.
Cover one baking sheet with lightly oiled plastic wrap, and place in the refrigerator. Transfer the other to the oven. Bake 10 minutes; reduce oven heat to 350 degrees. Bake 15 minutes more, until golden brown. Turn off oven; prop door open slightly to let steam escape. Allow gougeres to dry in the oven until crisp, about 10 minutes. Transfer to a wire rack. Raise heat back to 425 degrees. Repeat process for remaining batch. Serve warm.
Store in an airtight container in the refrigerator up to 3 days; reheat in a 350 degree oven for five minutes before serving.