Preheat oven to 425 degrees F. In a medium saucepan, combine the butter, salt, pepper, nutmeg, and water. Bring to a boil over medium-high heat. When the butter melts, reduce the heat to low. Add the flour to the butter-water mixture all at once, and cook over low heat, beating with a wooden spoon for 1 to 2 minutes, until the mixture pulls away from the sides of the pan. Remove pan from heat. Add the cheese and beat with the wooden spoon until well incorporated. Add 4 of the eggs, 1 at a time, beating each egg into the batter with the wooden spoon until thoroughly absorbed. Continue beating the mixture until it is smooth, shiny, and firm.
Drop the batter by small spoonfuls onto a lightly greased cookie sheet. Beat the remaining egg with ½ tablespoon water, then brush it onto the tops of the uncooked Gougeres. Bake in the upper third of the oven for 15 to 20 minutes, or until golden and doubled in size. Remove from the oven and serve hot, or allow to cool to room temperature.