Preheat the oven to 425℉, and prepare a sheet pan with parchment paper or a silicone baking mat.
Combine the butter with the milk, water, sugar, and salt in a pot and bring to a boil.
Remove from heat and dump in the flour all at once. Mix until fully combined with no lumps and return to medium-high heat.
Cook while stirring continuously for 3-5 or until a thin film forms at the bottom of the pot. If using a non-stick pot, oil droplets should form on the bottom of the pot.
Transfer dough to a heat resistance bowl and allow to cool to below 165℉.
Add in the first 3 eggs one by one, mixing until fully incorporated before adding the third egg.
Before adding the last egg, check for signs of proper moisture addition (read my above tell tale signs) and determine if you need it. If you do, scramble the egg, add half, determine if enough moisture has been added before adding the rest or more.
Transfer to a piping bag with a straight piping tip, and pipe round balls roughly 2" in diameter and 1.5" tall.
Sprinkle or spray light with water and push down any spiky peaks with a wet finger.
Bake for 15 minutes then reduce the temperature to 350℉ for the next 20 minutes. DO NOT open the oven during this time.
Turn off the oven and remove the pate a choux. Prick a small hole in the top for steam to escape and return to the oven while it's off and still warm for 30 more minutes. Remove from the oven and allow to fully cool.