Preheat the oven to 425F. Spray a large cookie sheet with cooking spray then line with parchment paper.
Whisk together the eggs and egg white into a measuring cup. You should have ½ cup. Discard any excess egg mixture.
Place the butter, milk, water, sugar and salt into a small saucepan over medium heat.
When the mixture comes to a boil, remove the pan from the heat, and stir in the flour. Keep stirring until the mixture clears the side of the pan.
Place the pan over low heat, stirring and smearing the mixture, until it is slightly shiny and small beads of fat appear on the bottom of the pan, about 2 minutes.
Transfer the dough to a food processor. Let cool until the temperature reaches 160F. Alternatively, process for 15 seconds with the feed tube open to cool it down.
With the processor running, pour in the egg mixture in a slow and steady stream.
Scrape down the bowl with a spatula, then process the mixture again for an additional 30 seconds. The pastry should be smooth, thick and sticky.
Transfer the pastry to a piping bag fitted with a French star attachment (Ateco 869).
Use a bench scraper to scrape the pastry towards the piping tip.
Pipe the paste into 12 - 5"x1" strips, about 1" apart on the cookie sheet.
Dip your finger into a bowl of cold water to even out and clean up the eclair shape.
Bake for 15 minutes. Reduce the temp to 375 and bake for an additional 10 minutes, until the eclairs are golden brown. Do not open up the oven until the eclairs are baked.
Cool completely. The eclairs should sound hollow when tapped.