Preheat the oven to 450 degrees.

Line two rimmed baking sheets with parchment paper and set aside.

In a medium heavy saucepan over medium-high heat, combine the water, milk, butter, salt and pepper and bring to a boil, stirring occasionally.

Once the mixture is boiling, reduce the heat to medium-low, add the flour all at one and continue to cook, stirring continuously with a wooden spoon.

Cook for about two minutes, or until the dough starts to form a cohesive ball and pull away from the sides of the pan. It should look like mashed potatoes and start smelling a bit nutty.

Remove the pan from the heat and transfer the dough to the bowl of a stand mixer with a paddle attachment. Let it sit and cool for 2 to 3 minutes, then beat the four whole eggs into the dough, one at a time, mixing completely between each addition.

Scrape down the sides of the bowl as needed, and continue to beat on medium speed for several more minutes until the dough is glossy and lemon-colored. Add the cheese and nutmeg and mix until combined. The dough will be sticky.

Fill a pastry bag fitted with a ½ inch round tip and pipe mounds of dough about an inch in diameter using about a Tablespoon of dough each, leaving 2 inches between each mound. Whisk the remaining egg yolk with one Tablespoon of water and brush the tops of each gougere.

Put the baking sheets in the oven and immediately reduce the heat to 350 degrees. Bake for 12 to 14 minutes or until the gougeres are puffy, golden brown, and sound hollow when you tap your fingertip on them.

Serve your gougeres warm from the oven or let them cool to room temperature. They're best eaten the same day they're baked but will store at room temperature for a few days, then briefly reheated to crisp them back up, or leftovers can be frozen, then defrosted and briefly reheated.