Preheat oven to 375℉. Line baking sheets with parchment paper or silpat mat. Set side.
In a medium saucepan, add water (or milk), butter, sugar, and salt. Bring to a boil over medium heat. Add flour and stir to combine using a wooden spoon or sturdy spatula. Bring to a boil as you stir. Continue to stir constantly until the dough starts to pull away from the sides of the saucepan.
Once dough is a smooth, cohesive ball, transfer to the bowl of stand mixer fitted with a paddle attachment. Beat a medium speed until the dough has cooled to just a little warmer than room temperature. *Note: The dough will steam as you beat it. Once dough is ready, add the whole eggs one at time. Make sure the egg is incorporated before adding the next one. Scrape down the bowl as needed. Once all the whole eggs are in, add the egg white and mix to combine. Continue to beat mixture until smooth, thick, and shiny. Use "V-test" to determine if dough is ready for piping.
Transfer batter to a pipping bag fitted with a 18mm plain round tip or star/fluted tip. Pipe 4-inch long cylinders on the lined baking sheet, at least 1 inch apart.
Bake in preheated oven for 20 minutes. Leave eclairs undisturbed during this time. Then, lower oven temperature to 325℉ and bake for another 20-30 minutes until eclairs are dry and feel very light in weight. Transfer to wire rack to let cool.