Preheat the oven to 400 °F.
Mix butter, water, whole milk, salt, and sugar in a pot. Bring to a simmer.
Add flour as you stir. Stir for 1 minute while the dough turns into a ball.
Move the dough into a bowl then allow it to cool down a few minutes.
In a stand mixer, slowly mix the mixture and add in the beaten eggs one at a time. Stop adding eggs as soon as it reaches a pipeable consistency. Look for a paste-like consistency in which the peaks are wet but barely lose their shape after you stop mixing. Use any leftover egg for the eggwash. See Note 1.
Line a baking sheet with parchment paper. Lightly brush the parchment paper with water.
Pipe 2-inch mounds onto the parchment paper, leaving about 2-3 inches of space between each mound. Gently smooth down peaks with the back of a spoon. See Note 2.
Use a baster to brush the tops with egg wash.