Preheat the oven to 425ºF (220ºC). Line 3 sheet pans with parchment paper.
Add the water, milk, and butter to a 2-quart pot over medium-high heat. Cook until the butter is completely melted.
Add the flour, and reduce the heat to low. Stir until the flour is absorbed, the mixture dries out a bit, and the dough pulls away from the sides of the pot.
Transfer the dough to a deep bowl, and set it aside to cool for 1 minute. Add one egg to the dough, and beat it with wooden spoon until fully mixed in (it’ll be loose at first, then thicken again). Repeat with the remaining eggs, beating them in one at a time.
Stir in 4 ounces of the cheese, pepper, and nutmeg.
Scoop the dough into a pastry bag fitted with a ½-inch round tip (or use a sturdy ziptop bag, and snip off the edge of one corner). Pipe 1-inch mounds of dough onto the prepared pans, leaving 1 1/2 to 2-inches between each mound. Sprinkle the tops with the remaining cheese
Bake, one tray at a time, 14 to 16 minutes, until golden and crisp. Pierce the cooked gougères with a metal skewer to release hot air so they don’t get soggy inside. Serve immediately, or let them finish cooling on the tray and store in a tin in cool, dry place until serving (they’ll keep for up to one day).