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.
Add all the flour at once and stir hard with a wooden spoon until all the flour is incorporated, 30 to 60 seconds. Return to the heat and cook, stirring for about 60 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 dough to a pastry bag fitted with a large round tip and pipe dollops roughly 1.5-2 Inches in diameter and an inch high. Lightly wet a finger and tap down the spike on top.
Egg Wash: In a bowl, whisk the egg and water together. Brush the surface of each with the egg wash.
Bake for about 20 minutes at 425F. Try not to open the oven door during the bake. Poke holes in the puffs and return to oven to bake an extra 2-3 minutes. Let cool on the baking sheet. You can leave the puffs in the oven with the door cracked open after baking if you'd like them to really dry out.