Preheat oven to 350°F. Line baking sheet with parchment paper or reusable silicone mat. Set aside.
In a medium sauce pot, mix together butter, water, milk and salt. Over medium heat, cook until butter melts and mixture has come to a boil. Add flour all at once and begin stirring with a wooden spoon. Continue to stir until dough pulls away from the sides of the pot. At this point the dough should be smooth.
Transfer dough to the bowl of a stand mixer fitted with a paddle attachment. On low speed, add eggs one at time. Make sure egg is fully incorporated before adding the next egg. As you add the eggs, the dough will separate. This is normal. Once all the eggs have been added, as you continue to mix, the dough will come back together. The dough should be shiny, smooth, and quite thick at this point.
Add cheese and black pepper. Fold in with a rubber spatula. Use a size 24 scoop to portion out batter. Space mounds at least 1 ½-inches apart on parchment lined baking sheet.
Brush each mound with egg wash and top with grated cheese. Bake in preheated oven for 30-35 minutes until the pastries have puffed up and have browned in color. The finished gougere should feel very light in weight for their size. If they feel heavy, put them bake in the oven for a couple of minutes.