In a medium saucepan over medium heat, mix together water, milk, butter and salt. Increase the temperature slightly and bring to a boil, stirring frequently.
Remove from heat and stir in 1 1/4 cup flour all at once using a wooden spoon. Once incorporated, place the saucepan back over medium heat and stir constantly for another 2 minutes to release extra moisture and partially cook the flour. The dough will look smooth and a thin film will form on the bottom of the pan.
Transfer the dough to a large mixing bowl and use an electric mixer to beat on medium speed for 1 minute to cool the mixture slightly. Add the 4 eggs 1 at a time, allowing eggs to fully incorporate between each addition. Once all eggs are incorporated, beat another minute until the dough is smooth and forms a thick ribbon when you pull up on the whisk.
Preheat oven to 400˚F and line baking sheets with parchment paper or Silpat mats. Transfer dough to a piping bag with a 1/2-inch opening, or use a large zip bag and cut a 1/2-inch opening at the tip.
Pipe out circular shapes (as small or as large as desired) onto prepared mats. Bake on middle rack for 15-20 minutes, until puffed up and golden brown. Let cool completely on a wire rack then slice in half with a serrated knife to prepare for the filling.