Preheat the oven to 425F. 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 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.
Using a pastry bag fitted with a large plain tip, pipe donut and dollops shapes.
Egg Wash: In a bowl, whisk the egg and water together. Brush the surface of each with the egg wash. Use your fingers to smooth out any bumps of points of dough that remain on the surface.
Bake for 15 minutes. Reduce the heat to 375 degrees and bake until puffed up and light golden brown, about 15 minutes more. Try not to open the oven door too often during the baking. Let cool on the baking sheet.
Using a pastry bag fitted with a 230 tip, gently pipe custard into each pastry, using only just enough to fill the inside (don't stuff them full).