Heat the oven to 425°F and adjust an oven rack to the middle. Line two half sheet pans with parchment paper.
Place the water, butter, and salt in a 3-quart saucier or saucepan, set over high heat, and bring to a boil. As soon as it boils, add all of the flour at once and stir with a wooden spoon until the mixture starts to come together, about 1 minute. Decrease the heat to low and continue stirring until the mixture forms a ball and is no longer sticky, 3 to 4 minutes.
Transfer the mixture to the bowl of a stand mixer fitted with the paddle attachment and mix on the stir setting until there is no more steam rising from the bowl, about 5 minutes. Increase the speed to low, then add the eggs, one at a time, making sure each is completely incorporated before adding another. You may need to stop the mixer occasionally and scrape down the sides of the bowl and the paddle every 2 additions of egg. Add one of the egg whites and mix until completely incorporated. Now check the mixture for consistency: It should tear slightly as it falls from the beater, creating a “V” shape. If it is still too stiff to do so, beat in the last white.
Add the Gruyère and cheddar and continue to beat just until combined, about 15 seconds.
Transfer to a piping bag fitted with a plastic coupler and a round tip. Pipe a little of the mixture into the four corners under the parchment paper to hold it in place on the sheet pans.
Pipe the mixture into 3/4- to 1-inch tall and 1 to 1 1/4 inches in diameter puffs on the parchment, 1 inch apart. You should be able to fit 30 rounds per tray. Use a damp finger to smooth the tops of the dough mixture. Sprinkle with the Parmesan.
Place one sheet pan of gougeres in the oven and bake for 15 minutes. Reduce the oven temperature to 350°F and continue to bake until golden brown and crisp, 8 to 10 more minutes.
Remove from the oven and immediately pierce the bottom of each puff with a paring knife to release steam. Increase the oven temperature back to 425°F and repeat the baking process with the second sheet pan of gougeres. Let cool to room temperature before serving.
Baked gougeres can be stored in zip-top bags and frozen for up to 1 month. If you’d like to reheat them, heat room temperature gougeres in a 325°F oven for 5 minutes. Reheat frozen gougeres directly from the freezer in a 325°F oven for 5 to 7 minutes.