Put the water, butter, salt in a heavy saucepan and bring to a boil over high heat. Add the flour and stir with a wooden spoon until the ingredients come together into smooth, uniform dough, about 2 minutes.
Transfer the mixture to the bowl of a stand mixer fitted with the paddle attachment. (You can also use a hand mixer.) Start paddling on low speed, then add the eggs, one at a time, until thoroughly absorbed, mixing for 1 minute between eggs, and stopping the motor periodically to scrape down the sides and bottom of the bowl with a rubber spatula. Finish with the final egg and mix for an additional 2 minutes.
Use the dough immediately. It does not refrigerate well.