Preheat oven to 375 F
Line two baking sheets with parchment paper
Combine the butter, water, milk, salt, and sugar 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 in once step. Stir until the flour is completely incorporated and a thick dough clumps into a ball. Press the ball against pan to "cook" the flour for about 30 seconds.
Remove from heat and transfer to the bowl of a stand mixer fitted with a paddle attachment Allow to cool down for a few minutes before adding the eggs in the next step.
With the mixer running on low or medium-low speed, slowly add the eggs in a few 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. You may not need to use all 4 eggs - you want a thick pipe-able consistency that is smooth and pliable (last time I made these I needed all 4 eggs)
Use your choux immediately or cover and refrigerate up to 3 days
Place your choux in a piping bag fitted with a round open tip nozzle
Wet your parchment lined sheet trays with a pastry brush
Pipe small 1-2 inch mounds of choux. I like to use a sharp small knife to cut the choux from the pastry tip (it's sticky)
Use a dampened finger to press down any peaks on your choux buns
Egg wash each choux mound
Bake for 30-40 minutes or until golden brown. During the last 5-10 minutes of the bake (but after 30) prick each bun with a toothpick or sharp knife to allow to dry inside. Do NOT open the oven before 30 minutes or your choux may collapse
Remove from oven or open oven door and allow to cool inside oven until room temperature
Split your buns for the vanilla whipped cream (or leave whole and pipe in pastry cream)