Preheat the Oven - Set the oven racks to upper-middle and lower-middle positions. Preheat to 425°F (218ºC). Line two sheet pans with parchment paper and set them aside.
Make the Dough - In a large heavy-bottomed saucepan, add 1 cup of water, sugar, salt, and butter. Bring to a boil over medium-high heat and stir until the butter fully melts, about 2 minutes.
Turn off the heat and immediately add the flour. Vigorously stir with a large spoon (non-metal) until the flour is incorporated. Increase the heat to medium, and constantly stir until it clumps together, about 4 to 5 minutes.
The dough should look relatively dry and just begin to leave a film on the saucepan.
Cool the Dough - Transfer the dough to a bowl of a stand mixer fitted with a paddle. Slowly stir on low speed (setting 2) to cool to 130°F (54ºC) or just below, about 2 to 3 minutes.
Add the Eggs in Stages - Add eggs one at a time, mixing on medium-low speed (setting 4) until each egg is fully incorporated, about 30 to 45 seconds per egg.
Once 3 eggs are added, check to see if the dough pulls away from the bowl's sides in thick threads, it will not clear the bowl.
It should be shiny but firm and not runny, so it’s easy to control when piped. If it’s still too thick, add the fourth egg and mix for 45 seconds.
Piping the Dough - Add a large plain tip to a piping bag. Lightly grease the inside of the pastry bag with cooking spray.
Add the dough to the pastry bag. Pipe onto the sheet pan, creating a 2-inch circular ball.
Pipe at least 2 inches apart from each other. They will expand to about 3 inches after baking.
Brush on Egg Wash - In a small bowl, whisk together one egg and 1 tablespoon of water. Brush the tops and sides of each dough ball with the egg wash.
Gradual Baking Process - Place the sheet trays in the upper-middle and lower-middle positions. Bake for 10 minutes at each temperature setting: 425°F (218ºC), 375°F (191ºC), 325°F (163ºC), 275°F (135ºC), 225°F (107ºC), 200°F (93ºC).
Do not open the oven door! It will take about 60 to 70 minutes of total baking time.
To check for doneness, break open one of the pastries. It should be as dry and feel light in weight.
If needed, continue to bake the pâte à choux at 200°F (93ºC). Transfer shells to a cooling rack. Cool completely and reserve until ready to fill.
Make the Whipped Cream - In a stand mixer fitted with the whisk attachment, add heavy cream, powdered sugar, and 1 teaspoon vanilla extract to the bowl.
Whip on medium-low speed (setting 4) until the cream is frothy with bubbles on the surface, about 1 minute.
Increase the speed to medium-high (setting 8), and whip until smooth, stiff peaks form, about 2 minutes.
If needed, continue to whip in 5-second intervals. Do not overwhip, or the cream will become curdled.
Piping the Cream - Working in batches, add the whipped cream to a pastry bag fitted with a large star tip. Use a serrated knife to cut the shells in half to create a top and bottom.
Evenly pipe the whipped cream into the pastry bottoms. Gently place the lids on top of the cream.
Top with Sugar - Sprinkle powdered sugar on top of each cream puff. Serve immediately, or refrigerate, uncovered, if not eating within one hour.