Preheat oven to 400 degrees. Heat 1 cup water, the butter, sugar, and salt in a medium saucepan over medium-high heat, stirring occasionally, until butter has melted and mixture is boiling. Remove from heat. Using a wooden spoon, stir in flour. Return to medium heat; cook, stirring constantly, until mixture pulls away from sides and a film forms on bottom of pan, about 4 1/2 minutes.
Transfer mixture to the bowl of an electric mixer fitted with the paddle attachment. Mix on low speed until slightly cooled, about 2 minutes. Raise speed to medium. Add 4 eggs, 1 at a time, mixing until incorporated after each addition. Batter should be shiny. Test batter by touching it with your finger and lifting to form a string. If a string does not form, mix in another egg a little at a time. If a string still doesn't form, add water 1 teaspoon at a time until it does.
Transfer to a pastry bag fitted with a 1/2-inch plain round tip (such as Ateco #806). Pipe 1-inch rounds onto baking sheets lined with parchment paper, spacing 1 inch apart. Whisk remaining egg and 1 teaspoon water in a small bowl; brush over tops of rounds
Bake until rounds are puffed and pale golden, about 10 minutes. Reduce oven temperature to 350 degrees. bake until golden, 20 to 25 minutes more. Turn off oven, and prop open oven door with a wooden spoon to release steam; let puffs dry 15 minutes. Using a spatula, transfer to a wire rack; let cool completely. Puffs can be refrigerated in an airtight container up to 1 day or frozen up to 1 month.