Preheat the oven to 425 degrees. Line a sheet pan with parchment paper.
In a medium saucepan bring the water, salt, sugar and butter to a rolling boil.
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 for about 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 4 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.
Pipe 1 to 1.5 inch dollops onto the parchment paper-lined baking sheet. Tap down the points with a wet finger. Brush lightly with an egg wash and then sprinkle with the Pecorino Romano.
Bake at 425F for 15 minutes. Or until golden brown. If your puffs need additional time, lower oven temp to 375F. Pierce the bottoms with a skewer and allow to cool upside down.
Once cooled, poke a hole into the bottom of each puff and then pipe in the filling.