Preheat oven to 375 degrees. Line two baking sheets with parchment paper. Set aside.
For the egg wash, whisk together 1 egg with 1 tablespoon water in a small bowl.
In a medium saucepan, combine 1 cup water with the butter, salt, and sugar over medium-high heat. Bring mixture to a boil, and immediately remove from heat. Using a wooden spoon, add flour, stirring vigorously, until flour is no longer visible. Return pan to medium-high heat. Cook, stirring constantly, until mixture pulls away from the sides of the pan and forms a film on the bottom, about 4 minutes.⏱ 4 min
Remove mixture from heat and transfer contents to a bowl. Add both cheeses, the herbs, and spices. Stir to combine with a wooden spoon and let cool slightly, about 3 minutes. Add remaining eggs, one at a time, stirring vigorously after each addition, and waiting to add the next egg until the previous one is entirely incorporated.⏱ 3 min
Transfer dough to a pastry bag fitted with a 1/2-inch round tip; pipe 1 1/2-inch puffs about 1 inch apart. Using a lightly moistened finger, smooth tops. Brush each puff with egg wash and sprinkle each one with remaining Gruyere.
Bake, rotating sheets halfway through, until puffs are golden, about 20 to 25 minutes. Serve immediately, as gougeres are best eaten warm from the oven.⏱ 23 min