Preheat oven to 450 degrees. Line 2 baking sheets with parchment paper.
Combine Champagne, 1/2 cup water, butter, salt, and sugar in a medium saucepan. Bring to a boil over medium heat. Immediately add flour, and stir vigorously with a wooden spoon until the mixture pulls away from the sides of the pan and a film forms on the bottom, 1 to 2 minutes.
Transfer mixture to the bowl of a food processor. Mix 2 minutes to allow steam to escape. Then add 4 eggs, one at a time, completely incorporating each one before adding the next. Add cheese, and continue mixing batter until well combined, 1 to 2 minutes. Transfer to a piping bag fitted with a 1/2-inch plain tip (such as Ateco #807), and pipe into 1-inch rounds on the prepared baking sheets. Beat remaining egg and with 1 teaspoon water in a small bowl; brush over each round of dough.
Bake gougeres 15 minutes, then turn oven down to 350 degrees and continue to bake until puffed and golden brown, 5 to 8 minutes.