These gougères turned out light as air, flaky, and moist, and most of all they all had a lovely hollowed center, the signature of this savory version of pâte à choux!
Place the egg in a small bowl. Add the water and whisk until well blended.
Preheat the oven to 375 degrees.
Sift the flour in a medium-sized bowl if using a handheld mixer or into the bowl of an electric mixer. Make a well in the center and set aside.
Place the milk, water, butter, salt, and cayenne in a medium-sized saucepan. Bring to a boil over medium-high heat. As soon as the liquid boils, remove the pan from the heat and add to the flour, pouring it in the center of the well. Whisk on medium speed until the mixture is thick and forms a smooth, doughy batter.
Transfer the batter back to the pan (the one you used to boil the milk mixture). Heat over medium heat, working it with a wooden spoon until it dries up a bit and comes together, about 2-3 minutes.
Transfer back into the whisking bowl. Add 1 egg and whisk with the electric mixer on medium speed until well incorporated. Repeat, adding the eggs 1 at a time, and whisking each time until well incorporated.
Add the cheese to the batter and whisk on medium speed until the cheese has melted and the batter is smooth, about 30 seconds. Scrape down the sides of the bowl with a silicone spatula and shape the batter into a round ball.
Line 2 baking sheets with Silpat or parchment paper. Transfer the batter to a piping bag fitted with a ½-inch smooth tip. Pipe 1 ½-inch diameter mounds, 2 inches apart, onto the baking sheets. Brush with the egg wash and bake until the tops are golden, for 22-25 minutes. Make sure not to undercook the gougères or they will collapse when they cool. Serve immediately, while piping hot.