Preheat the oven to 450°F (232°C) and line two baking sheets with parchment paper. In a medium saucepan over medium-high heat, combine the butter, milk, Dijon mustard, cayenne pepper, and water. When the mixture comes to a boil, reduce the heat to medium-low and add the flour. Cook, stirring constantly, until the mixture forms a ball and dries out slightly, about 2–3 minutes.
Transfer the flour mixture to the bowl of a stand mixer and beat it with the paddle attachment for 1 minute. Add the eggs one at a time, beating until fully incorporated before adding the next. Once all the eggs are incorporated, stir in the cheese and scallions, mixing well.
Drop the batter by a scant tablespoonful onto the baking sheets, leaving 1 inch (2.5 cm) between them. Bake for 7 minutes. Then, reduce the oven temperature to 350°F (180°C) and continue baking until the gougères are golden brown, about 15 to 20 minutes. Immediately after removing from the oven, poke each puff once with a toothpick, sharp chopstick, or skewer to allow steam to escape. Serve immediately.
Notes
If you use unsalted butter, add 1/2 teaspoon of salt.