How to Make It
Preheat the oven to 400°. Line 2 baking sheets with parchment paper. In a large saucepan, combine the milk with the water, butter and salt; bring to a boil over moderately high heat. Add the flour all at once with the Piment d'Espelette and beat vigorously with a wooden spoon until the flour is thoroughly incorporated. Reduce the heat to low, return the saucepan to the burner and cook the gougère dough, stirring constantly, until the dough pulls away from the side of the pan, about 3 minutes.
Remove the saucepan from the heat and let stand at room temperature, stirring occasionally, until the dough cools slightly, about 5 minutes. Add the eggs one at a time, stirring briskly between additions to thoroughly incorporate each egg.
Drop 3-tablespoon mounds of dough onto the baking sheets, 2 inches apart. Top each round with 1 tablespoon of cheese; sprinkle with fleur de sel and pepper.
Bake the gougères for 15 minutes. Reduce the oven temperature to 350° and bake for 30 minutes longer, switching the baking sheets halfway through, until the gougères are puffed and browned. Turn off the oven, propping the door ajar with a wooden spoon. Let the gougères rest in the oven for about 30 minutes longer, until crisp on the outside but still steamy within. Serve warm or at room temperature.