1. Preheat oven to 400 degrees. Heat water, butter, sugar, and salt in a medium saucepan over medium-high heat, stirring occasionally until butter is melted and mixture is boiling. Remove from heat. Using a wooden spoon, stir in flour. Return to medium heat, and cook, stirring constantly, until mixture pulls away from the sides and a film forms on the bottom of the pan, about 4 minutes.

2. Transfer to the bowl of a mixer, and beat on low speed until slightly cooled, about 2 minutes. Increase speed to medium, and add 2 eggs, one at a time, letting each incorporate before adding the next. Beat until dough is smooth and shiny, about 1 minute more. Test the dough by touching it with your finger and lifting to form a string. If string doesn't form, add water, 1 teaspoon at a time, until it does.

3. Transfer dough to a pastry bag fitted with a 1/2-inch plain tip. Pipe small dots at the corners of baking sheets, and lay parchment on top, pressing to secure. Pipe 1-inch rounds onto parchment, spacing 1 inch apart. Brush with egg wash, and sprinkle with Parmesan, salt, and pepper.

4. Bake until rounds are puffed and pale golden, about 10 minutes. Reduce oven temperature to 350 degrees, and bake until golden and dry in the center, 20 to 25 minutes.