In a medium saucepan set over medium heat, bring the water, Champagne, butter, and salt to a simmer, whisking until the butter melts (no need to let the mixture come to a full boil). Add the flour all at once; stir vigorously with a wooden spoon until the flour absorbs all the liquid and the dough forms a ball, pulling away from the sides of the pan (this should take 30 seconds to a minute). Keep on stirring vigorously over the heat until a film forms on the bottom of the pan and the dough is no longer sticky, 1 to 2 minutes longer.
Remove the pan from the heat and let the dough cool (in the pan) for 2 to 3 minutes.
Transfer the dough to the bowl of a stand mixer, or to a large mixing bowl, if using a hand mixer. Beat the eggs into the warm dough, one at a time. After each addition, the mixture will first look curdled but it will come back together upon beating. Make sure each egg is well incorporated before adding the next.
Once all the eggs are incorporated, the dough will be thick but creamy. Fold in the Parmesan cheese, thyme, and some freshly ground black pepper.
To portion the parmesan cheese puffs, you can transfer the dough to a pastry bag fitted with a large, round piping tip, or—the easiest and fastest method—a small 0.5-oz (1 tbsp) cookie scoop, which creates perfectly round shapes. (If you use a larger cookie scoop, make sure to increase the baking time accordingly.) Drop each mount of dough about 3 inches (7 cm) apart on the parchment paper-lined baking sheets. The dough should be thick enough to keep its round shape. If desired, use damp fingertips to press down any peaks of dough. (This will create rounder puffs.)
The parmesan cheese puffs will not all fit on one single baking sheet; refrigerate the remaining dough while the first batch of cheese puffs bakes. (See note for freezing tips.)
Using a pastry brush, lightly brush the portioned dough with the egg wash. Bake the cheese puffs until golden brown, about 22 minutes. The parmesan cheese puffs should seem firm, but inside, they should be slightly eggy and moist.
Transfer the parmesan cheese puffs to a wire rack. (Cheese puffs will deflate slightly as they rest. If the cheese puffs deflate a lot, they were probably underbaked. Add a few minutes of baking time for the next batch.)
SERVING: Serve the parmesan cheese puffs hot or warm. Cheese puffs can be made several hours ahead. If making ahead, let the cheese puffs cool completely on the wire rack, then store in an airtight container at room temperature for up to 24 hours. Before serving, rewarm in a 325°F (160°C) oven for 5 to 10 minutes.
NOTE—How to freeze raw parmesan cheese puffs: portion all the dough, setting the raw cheese puffs close together on a parchment paper-lined baking sheet, then freeze until hard. Transfer the frozen dough to an airtight freezer bag or container and keep frozen for up to 1 month. You can bake the cheese puffs from their frozen state: simply add about 5 to 8 minutes to the baking time provided to make sure they are cooked through.
No need to open a bottle of French champagne to pour a half cup for this recipe—although if you know you will pair the rest of the bottle with the gougère, by all means, indulge! You can substitute any dry sparkling wine such as Cava, Prosecco, other Champagne-style wines, or even hard cider.
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