Combine butter, water, milk, salt, and granulated sugar in a medium saucepan over medium heat, stirring until the butter has melted. Bring the mixture to a simmer and then reduce heat to low. Add all of the flour and stir until completely incorporated. The dough should be a thick ball. Press against bottom of sides of the pan for 1 minute and remove from heat.
Place ball of dough into a large mixing bowl and allow to cool before adding eggs.
With hand or stand mixer on low speed, slowly add the beaten eggs in 3 or 4 batches with 30 seconds between each. Pour in very slowly and observe the texture of the dough. At first it won't look like it's coming together, but will eventually become thick and glossy. Stop when you have a thick and pipeable texture. You may have a bit of beaten egg left over.
Preheat oven to 400 degrees F. Line two baking sheet with parchment and lightly brush each sheet with water.
Fill your piping bag with 1/2 in. tip or Ziploc bag with the choux pastry dough. Pipe 3 in. strips about 3 inches apart. Moisten your finger with water and smooth down any points.
Bake for 20 minutes and then, making sure not to open the oven door, reduce oven temperature to 350 degrees F and bake for 10-15 more minutes until golden brown. Remove from oven and transfer to a cooling rack. Cool completely before filling.