Heat the oven to 425°F and adjust oven rack to the middle. Line two half sheet pans with parchment paper.
Place the water, butter, and salt in a 3-quart saucier, set over high heat, and bring to a boil.
As soon as it boils, add all of the flour at once and stir with a wooden spoon until the mixture starts to come together, about 1 minute.
Decrease the heat to low and continue stirring until the mixture forms a ball and is no longer sticky, 3 to 4 minutes.
Transfer the mixture to the bowl of a stand mixer fitted with the paddle attachment and mix on low speed for 5 minutes to cool, or until there is no more steam rising.
With the mixer still on low speed, add the eggs and egg whites, one at a time, making sure each is completely incorporated before adding another. You will need to stop the mixer occasionally and scrape down the sides of the bowl and the paddle attachment.
Before adding the last egg white, check the mixture for consistency: It should tear slightly as it falls from the beater, creating a“V” shape. The mixture may only need 1 egg white.
Transfer to a piping bag fitted with a plastic coupler and a round tip. The dough will puff up to approx twice the height, so pick your piping tip based on this.
Pipe a little of the mixture into the four corners under the parchment paper to hold it in place on the sheet pans.
Pipe the mixture on to the parchment paper (see below). Use a clean dampened finger to smooth any tips left from piping.
Place the sheet pan in the oven and bake for 15 minutes.
Reduce the oven temperature to 350°F and continue to bake until golden brown, 10 to 12 more minutes.
Remove from the oven and immediately pierce the bottom of each (eclair or ball) with a paring knife to release steam. Repeat with the remaining pastries.
Let cool completely before filling.
