Make egg wash: combine a little beaten egg with a little milk or water.
Combine the butter, water, milk, salt, and granulated sugar together in a medium saucepan over medium heat. Stir until the butter has melted. Bring mixture to a simmer. Once simmering, reduce heat to low and add the flour all at once. Stir until the flour is completely incorporated and a thick dough clumps into a ball. Mash the dough ball against the bottom and sides of the pan for 1-2 minutes, which gently cooks the flour. Remove from heat and transfer to the bowl of a stand mixer fitted with a paddle attachment or, if using a hand-held mixer, a large mixing bowl. Allow to cool down for a few minutes before adding the eggs in the next step.
With the mixer running on low speed, slowly add the eggs in 3-4 separate additions mixing for 30 seconds between each. The mixture will look curdled at first, but will begin to come together as the mixer runs. Pour in the final addition of beaten eggs very slowly. Stop adding when the choux pastry has reached the desired texture: shiny, thick, and smooth with a pipeable consistency. I usually leave a few teaspoons of beaten egg behind, which can be used with the egg wash.
Choux pastry dough can be used immediately or cover and refrigerate for up to 3 days.
For cream puff and profiterole shells:
Preheat oven to 400F. Line two baking sheets with parchment paper. Lightly brush the parchment with water, which creates a humid environment for the pastry shells allowing them to puff up without drying out or burning.
Transfer choux pastry dough to a piping bag fitted with a Wilton 1A piping tip (or zip-topped bag with the corner cut off). Pipe 2-inch mounds about 3 inches apart. Using a water moistened finger, smooth down the peaks and lightly brush each with egg wash.
Bake for 20 minutes then, keeping the pastries in the oven, reduce oven to 350 F and continue to bake for 10-15 more minutes until golden brown. Do not open the oven as the pastries cook, as cool air will prevent them from properly puffing up. Remove from the oven and transfer to a cooling rack.
Allow to cool completely before filling. Split open pastries and fill with homemade whipped cream, lemon curd, pastry cream, jam, or combination. Or poke a hole in the pastries and pipe the filling inside.