Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper and set aside.
Melt the butter in a medium saucepan over medium heat. Add the milk and heat until small bubbles start to form around the edges. Add the flour and cook, stirring constantly with a wooden spoon, until the mixture is smooth and glossy and a dough ball forms. (The dough should start to pull away from the edges of the pan.)
Transfer the dough to a stand mixer fitted with the paddle attachment. Mix in 5 of the eggs, one at a time, on medium speed until completely combined. Stir in the salt, pepper and 3/4 cup of the Gruyere.
Spoon large tablespoons of the batter onto the prepared baking sheet, leaving about 1 inch between each puff. Whisk the remaining egg and use it to brush the tops of the puffs. Sprinkle over the remaining 1/4 cup Gruyere.
Bake until puffed and golden, about 25 minutes. Serve hot.