Preheat oven to 425 degrees Fahrenheit. Line a baking sheet with parchment paper.
In a saucepan over medium-high heat, add milk, butter, salt, and cayenne pepper. Cook until the butter is melted. Reduce to low heat.
Add flour and stir until flour no longer sticks to the sides of the pan. Remove from the heat.
Scramble 3 of the eggs in a separate dish. Slowly stir the eggs into the flour mixture.
Add cheese and stir to combine.
Using a cookie scoop, add scoops of the cheese puff mixture to the pan.
Scramble the remaining egg and add 1 tablespoon of water. Brush the tops of all of the cheese puffs with the egg wash.
Bake in the oven at 425 degrees Fahrenheit for 10 minutes. Reduce heat to 350 degrees Fahrenheit and cook for an additional 18-20 minutes until they reach a golden brown.