Preheat your oven to 500˚F and line a sheet pan with parchment paper.
In a saucepan over low heat, melt butter, water, milk, salt, and pepper.
Bring it to a boil, then add all the flour, stirring vigorously. Remove from heat when fully combined.
Return to medium heat, cook for a few minutes until a thin film covers the pan’s bottom and the dough easily pulls away from the sides.
Transfer the dough to a bowl, spread it up the sides to cool.
Beat eggs in a separate bowl, then add to the dough gradually, beating with an electric mixer until glossy.
Stir in 2.5 oz of shredded cheddar cheese.
Transfer the dough to a piping bag with a ½-inch round tip.
Pipe 1 ½ tablespoons of dough onto parchment paper, flatten the tops, and sprinkle remaining cheddar on top.
Place in the 500˚F oven, turn off the oven, and set a timer for 15 minutes. Do NOT open the door.
After 15 minutes, turn the oven back on to 350˚F and bake for another 10 minutes. Optionally, stab each puff with a fork after 5 minutes for extra crispiness.