Preheat the oven to 425 degrees F. Line a baking sheet with parchment paper or brush it with butter and dust with flour.
  In a medium saucepan, bring the water, butter, and the salt to a boil. When the butter is melted, remove the pan from the heat and sift in the flour. With a wooden spoon, stir the flour and liquid together to make a paste. Return the pan to the heat, and stir the paste to dry it out; in about 2 minutes it will form a smooth, shiny ball that pulls away from the sides of the pan.
  Transfer the paste to a mixing bowl; stir with a wooden spoon for 1 minute to cool the paste.
  Whisk one egg in a small bowl and reserve 1 tablespoon of the whisked egg to use as a glaze for the puffs. Add the remainder of the whisked egg plus the other 2 eggs, one at a time, to the paste, making sure each egg is fully incorporated in the dough before adding the next egg. When doing this by hand, the egg may at first seem to resist mixing with the dough, but stay with it and the egg will bind with the paste to make a stiff dough. Once all the eggs are incorporated, mix in the cheese, mustard, nutmeg, and cayenne.
  With a pastry bag or 2 spoons, form the dough into plump 1-inch rounds, spaced 1 1/2 inches apart on the prepared baking sheet. If using spoons, dip them in warm water after forming each round so the dough doesn't stick to the spoons. If you are using a pastry bag, gently squeeze the bag to form the rounds; release the pressure before you lift the bag to avoid lopsided tails.
  Lightly brush the rounds with the reserved egg. Bake the cheese puffs for 20 minutes until they are puffed and golden brown. For crisp, light puffs, do not open the oven door while they are cooking. Serve warm. The gougeres can be made a day ahead and frozen for future use, then reheated in a 350 degrees F oven for 5 minutes.