Add the butter, water, milk, sugar, and salt to a medium saucepan. Then bring the mixture to a boil over medium heat while stirring often.
Remove the boiling wet mixture from the heat. Then add the flour all at once and stir it into the wet mixture with a wooden spoon until it’s fully incorporated and forms a compact ball.
Put the saucepan back on the stove and cook the mixture for 1 to 2 minutes over low heat while stirring constantly. When it’s done the dough will look smooth and there will be a film on the bottom of the pot.
Transfer the dough to the bowl of a stand mixer with a whisk attachment. If using an electric hand mixer, put the dough in a large bowl.
Allow the dough to cool down for a few minutes.
Beat the room-temperature eggs one at a time into the dough at low speed. Each egg should be fully incorporated before adding the next. Once all the eggs have been added, beat the dough for another minute until it’s a smooth, shiny, pipeable consistency.
Put the dough into a piping bag with a ½-inch opening. You can also use a large ziplock bag and cut a ½-inch opening at the tip.
Heat the vegetable oil to 375˚F in a deep pot. You can also use a deep fryer if you prefer. Next, pipe 1-inch lengths of the dough into the hot oil. You need to pipe it close to the surface and pull away quickly. Then let the donuts fry for about 2 minutes on one side and then flip them over to fry for 2 more minutes. When done cooking, transfer them to a paper towel-lined plate to drain the excess oil.
Generously dust the zeppole with powdered sugar and serve.