Preheat oven to 400°F. Line 2 large baking sheets with parchment paper. Bring butter and 1 cup water to a boil in a large heavy saucepan. Remove from heat. Add flour, cayenne, and salt; stir vigorously to blend. Return to heat; stir vigorously over medium heat until mixture forms a ball and a thin, dry film forms on bottom and sides of pan, about 1 minute. Transfer mixture to a stand mixer fitted with a paddle attachment. Add 1 cup Gruyère, Parmesan, and 2 teaspoons thyme; beat on low speed to blend. Add 4 eggs, one at a time, fully incorporating each egg between additions, and scraping down sides of bowl as needed.
Spoon dough into a pastry bag fitted with a 1/2" tip or a plastic bag with 1/2" cut diagonally from 1 corner. Pipe dough onto prepared baking sheets in 1"–1 1/4" rounds, spacing 1" apart. Whisk remaining egg with 1 tablespoon water in a small bowl. Brush egg wash lightly over rounds, patting down to form rounded tops. Sprinkle with remaining 3 tablespoons Gruyère. Garnish each with a few thyme leaves.
Bake gougères until puffed and golden brown and centers are cooked through but still moist, 20-25 minutes. Let cool slightly and serve.
Make ahead: Let cool completely. Store airtight in freezer. Rewarm in 325°F oven until hot, 10-15 minutes.