Heat the oven to 450°F. Line two baking sheets with foil or parchment paper, and lightly grease the lining. Combine the butter, milk, mustard, salt, cayenne pepper, and ½ cup water in a medium saucepan over medium-high heat. When the mixture comes to a boil, reduce the heat to medium-low and whisk in the flour. Cook, stirring constantly, until the mixture forms a ball and dries out slightly, about 3 minutes.
Transfer the flour mixture to the bowl of a stand mixer and beat it with the paddle attachment for 1 minute. Beat in the eggs, one at a time, waiting until each egg is fully incorporated into the dough before adding the next. When all the eggs are incorporated, stir in the cheese.
Drop the batter by the scant tablespoonful onto the baking sheets, leaving 1 inch between gougères. Bake at 450°F for 7 minutes. Reduce the oven temperature to 350°F and continue to bake until the gougères are golden brown, 18 to 20 minutes. Immediately after removing from the oven, poke each gougère once with a toothpick, sharp chopstick, or skewer to allow steam to escape. Serve hot, warm, or at room temperature.
