In a medium saucepan, combine 1/4 cup water, milk, butter, salt and pepper and bring to a boil over medium heat. Add the flour, and stir with a wooden spoon until the mixture becomes dry, about 1 minute.
Transfer the mixture to a large mixing bowl, add the cheese and beat with a hand mixer on medium speed for 1 minute. Add the eggs one at a time. Continue to mix on medium-high speed until the dough is smooth and shiny, 1 to 2 minutes.
Spoon the dough into a large resealable plastic bag and seal it shut. Cut a 1/4-inch diagonal slice from one corner of the bag; squeeze the dough through that hole and into 1-inch rounds on the baking sheet, leaving a 1-inch space between each gougère. Smooth any peaks by dipping your finger in water and very gently rubbing the top of each piped round.
Transfer the baking sheet to oven. After 10 minutes, rotate the baking sheet and reduce oven temperature to 350°. Continue to bake until gougères are golden brown, about 15 minutes more. Remove gougères from oven, transfer to a serving platter or basket, and serve immediately. Leftovers can be reheated by placing them in a 400° oven for 2 minutes.