Preheat the oven to 400 degrees F.
In a small mixing bowl, add the grated cheese and plenty of freshly cracked black pepper to the half-recipe of pate a choux. With a rubber spatula, scoop the pate a choux into the pastry bag and pipe out approximately 25 (1-inch) rounds, spaced 1 to 2 inches apart on the parchment paper-lined baking sheet. Brush lightly with the beaten egg and place in the oven. Cook until golden and puffed, about 25 minutes. Remove from the oven and cool briefly on a baking rack. Serve hot or at room temperature.
In a small saucepan over high heat, bring the water, salt, sugar, and butter to a boil, making sure the butter is completely melted. Off the heat, add the flour all at once and beat vigorously with a wooden spoon. Return to the heat and continue beating until the dough forms a solid, smooth mass and pulls away from the sides of the saucepan. Take off the heat and empty the dough into a clean mixing bowl. Little by little add the beaten eggs, beating vigorously in between each addition, until the dough forms a smooth, supple mass. Divide the dough into 2 even quantities, 1 part to be used for the gougeres, the other for profiteroles.