Preheat oven to 350 degrees F.
In a medium saucepot, 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, black pepper, and thyme. Fold in with a rubber spatula. Transfer dough to a pastry bag fitted with a 1/2 inch plain round tip. Pipe 1-inch mounds onto a parchment lined baking sheet. Space mounds at least 1 1/2 inches apart. Alternatively, you can use a spoon or a small ice cream scooper to make 1-inch rounds.
Brush each mound with egg wash (whisk together 1 egg and 1/8 tsp salt) and top with cheese. Bake in preheated oven for 20-30 minutes until the pastries have puffed up and have browned in color. The finished gougeres should feel very light in weight for their size. If they feel heavy, put them bake in the oven for a couple of minutes.
Tip: Place a little batter/dough on the corners of the sheet tray to keep the parchment in place while you pipe.
RECIPE ADAPTED FROM TARTINE