Step 1 Heat the oven to 450 degrees. Line 2 baking sheets with silicone baking sheets or parchment paper.
Step 2 In a medium saucepan, combine the water, butter, salt and sugar and bring to a boil over high heat. Add all the flour at once, reduce the heat to medium and stir with a wooden spoon for 2 minutes, or until the dough forms a ball and the excess moisture has evaporated (if the ball forms more quickly, continue to cook and stir for a full 2 minutes).
Step 3Transfer the mixture to the bowl of a stand mixer fitted with the paddle and beat for about 30 seconds at medium speed to cool slightly. Add 4 eggs and continue to mix until completely combined and the batter has a smooth, silky texture. Stop the machine and lift up the beater to check the consistency of the batter. It should form a peak with a tip that falls over. If it is too stiff, beat in the white of the remaining egg. Check again and, if necessary, add the yolk. Finally, mix in three-fourths cup of Gruyere and adjust the seasoning with salt and white pepper.
Step 4Fill a pastry bag fitted with a three-eighths-inch plain pastry tip with the gougeres batter. Pipe the batter into 1-tablespoon mounds on the baking sheets, leaving about 2 inches between the gougeres, as the mixture will spread during baking. Sprinkle the top of each gougere with about one-half teaspoon of the remaining grated cheese. (At this point, the gougeres can be frozen on the baking sheets until solid, then stored in the freezer in a tightly sealed container.)
Step 5When almost ready to serve, transfer the frozen gougeres back onto the lined baking sheet and bake at 450 degrees until they puff and hold their shape, 7 to 8 minutes. Reduce the heat to 350 degrees and bake until they are a light golden brown color, an additional 20 to 25 minutes. When you think the gougeres are done, break one open; it should be hollow and the inside should be cooked but still slightly moist.
Step 6Remove the pans from the oven and serve the gougeres while hot.