Heat oven to 375 degrees. Line baking sheets with nonstick baking mats; set aside.
In a small saucepan, bring 3/4 cup water, butter, and salt to a boil. Immediately remove from heat. Add flour all at once, and beat vigorously with a wooden spoon until the mixture pulls away from the sides of the pan and a film forms on the bottom, about 30 seconds.
Transfer to the bowl of an electric mixer fitted with a paddle attachment. Beat 4 eggs into the batter one at a time, on medium speed, letting each one incorporate completely before adding the next. Add the cheese, and continue beating batter until well combined. Transfer batter to a pastry bag fitted with a 1/2-inch plain tip (Ateco #9807) and pipe into 1-inch mounds on the prepared baking sheets. Beat the remaining egg; brush each mound with some of the egg to glaze. If a pointy tip has formed on top, gently use your finger to flatten it.
Bake until puffed, golden, and the center is slightly damp but no wet dough remains (test by cutting into the center of one), about 25 minutes. Let gougeres cool slightly before removing from baking sheet. Serve warm.