Preheat the oven to 425 degrees. Line a sheet pan with parchment paper.
In a large saucepan, bring the water, butter, salt and sugar to a rolling boil over medium-high heat. When it boils, immediately take the pan off the heat. Stirring with a wooden spoon, add all the flour at once and stir hard until all the flour is incorporated, 30 to 60 seconds. Return to the heat and cook, stirring, 30 seconds.
Scrape the mixture into a mixer fitted with a paddle attachment (or use a hand mixer). Mix at medium speed. With the mixer running, add 3 eggs, 1 egg at a time. Stop mixing after each addition to scrape down the sides of the bowl. Mix until the dough is smooth and glossy and the eggs are completely incorporated. The dough should be thick, but should fall slowly and steadily from the beaters when you lift them out of the bowl. If the dough is still clinging to the beaters, add the remaining 1 egg and mix until incorporated.
Transfer to a pastry bag fitted with a large round tip or an 869, pipe fat dots about 1 1/2 inches wide and an inch high spaced 2 inches apart onto a parchment-lined pan.
Dip your finger in water and press the tops to smooth out any points of dough that remain on the surface.
Whisk an egg and water together. Brush the surface of each eclair with the egg wash.