Preheat oven to 450° and line 2 pans with parchment paper or baking sheet liners.
In a large sauce pan, combine the water, milk, butter and salt and bring the mixture to a full boil over medium-high heat. When the butter has melted add the flour, all at once, and stir vigorously with a wooden spoon until the mixture forms a ball that pulls away from the sides of the pan.
Turn the heat down to medium and cook the dough for another 3 or 4 minutes to dry out and cook the flour taste out of the dough. It will look a little shiny and there will some starch remaining on the bottom of the pan and the dough will be stiff.
Transfer the dough to the bowl of a stand mixer fitted with a paddle attachment. (Alternatively, use an electric hand mixer or beat by hand with a stiff spatula.) Beat the dough on medium-low speed until it stops steaming and is just warm to the touch, about 1 minute.
Continue beating and add the eggs in one at a time. Wait for each egg to be absorbed completely before adding the next. Scrape down the sides of the bowl if needed. Once the dough comes together after the addition of all the eggs, add the cheese and mix until it’s well distributed.
Scoop rounded tablespoons of dough onto the baking sheets, spacing the about 1-inch apart. They can be piped on using a piping bag or a storage bag with the corner snipped off.
Press the top down with a fork or spoon dampened slightly with water, until the tops are smooth. Brush with the egg wash mix and bake for 5 minutes. Turn the heat down to 350° and bake for 20-25 minutes until they are golden brown. Rotate the sheets half way through the baking period. If the inside of the gougere isn’t done, turn off the oven and prop the door open with a wooden spoon for about ½ hour so they can dry out a little.
The gougere’s can be stored in the refrigerator up to a week or frozen for a month. You can fill these with any kind of cheese stuffing or serve them as a stand-alone bite.