Gougères are savory cheese bites made with steam-leavened dough called pate a choux, originally from France.
Preheat oven to 375 degrees. Combine1 cup water, butter, salt, and cayenne in a saucepan. Bring to a boil. When butter melts, add flour. Reduce heat to medium low; stir constantly with a wooden spoon until dough pulls away from edges. The dough should not stick to fingers and will appear shiny and smooth. Transfer to a medium bowl; let cool 5 minutes.
One at a time, add 4 eggs; mix well between additions. Add Parmesan and chives; mix until combined. Transfer to a pastry bag with a plain 1/2-inch round tip (Wilton #1A). To form each gougere, pipe 1 teaspoon of dough onto a parchment-lined baking sheet. Allow 2 inches between each gougere. Push down peaks of dough with a finger dipped in water.
Combine remaining egg with remaining tablespoon water; brush on gougeres. Bake 20 to 25 minutes, rotating pan after 12 minutes, until golden. Serve.