Preheat oven to 400°F and line two baking sheets with parchment paper.
Combine butter, water, salt, pepper and dill in a medium saucepan over high heat. When the mixture comes to a boil, remove from heat and add in the flour all at once. Immediately fold flour into the mixture until the flour disappears. Continue to cook over medium heat for 1-2 minutes and the dough becomes one large ball.
Transfer dough to a large bowl. Add one egg at a time, beating together fully on medium before adding the next egg. After the fourth egg is incorporated, mix in the cheese.
Transfer the dough to a large pastry bag fitted with a ½-inch round tip. Hold the pastry bag straight up and pipe 1 x 1-inch rounds on the prepared sheets, gently lifting the tip up as you pipe. If your rounds have defined peaks from the piping tip, gently tap down the peak with a water-moistened finger.
Whisk together egg yolk and 1 tsp water to create an egg wash. Gently brush over piped rounds.
Bake for 23-25 minutes until puffed and lightly browned, rotating midway through. The gougerès should be dry in the center and feel hollow to the touch. Let cool for 5 minutes and enjoy warm.
