Preheat oven to 400F and line a baking sheet with parchment paper or silpat.
Bring 1 cup water, the butter, salt, and cayenne to a boil in a heavy 4-quart saucepan; boil until butter has melted.
Remove from heat; stir in flour.
Return to burner; cook, stirring, until mixture comes together and pulls away from the pan, about 2 minutes more.
Remove from heat.
Using a wooden spoon, beat in four of the eggs, one at a time, until they are incorporated and batter is smooth.
Stir in zest and herbs.
Transfer mixture to a piping bag, and cut a 1/2-inch opening.
Pipe 1-inch rounds onto prepared baking sheets. (Alternatively, drop rounded teaspoons of batter onto baking sheets lined with parchment. If not using immediately, freeze up to 1 month in airtight containers or resealable plastic bags.).
Whisk together remaining egg and one teaspoon water in a small bowl; lightly brush on top of each puff.