Preheat the oven to oven to 450°F (232°C). Line two baking sheets with parchment paper. Combine the butter, milk, Dijon mustard, cayenne pepper, and water in a medium saucepan over medium-high heat.
When the mixture comes to a boil, reduce the heat to medium-low and add in 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. Beat in the eggs, one at a time until fully incorporated into the dough before adding the next. When all the eggs are incorporated, stir in the cheese and scallions, mix well.
Drop the batter by a scant tablespoonful onto the baking sheets, leaving 1 inch between them. Bake for 7 minutes. Reduce the oven temperature to 350°F and continue to bake 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.