Preheat the oven to 400°. Line 2 large baking sheets with parchment paper. In a medium saucepan, combine the water with the butter, cayenne, cup of the milk and 1/4 teaspoon of the salt and bring to a boil. Add the flour and baking powder and cook over moderate heat, stirring constantly with a wooden spoon, until the dough pulls away from the side of the pan, about 2 minutes.
Transfer the dough to the bowl of a standing mixer fitted with the paddle. Add 1 cup of the cheese and mix at medium speed until the dough cools slightly, about 1 minute. Add the eggs one at a time, mixing well between additions. After all the eggs have been added, mix the dough until shiny, about 5 minutes.
Using a pastry bag, pipe the dough into 1-inch mounds about 1/2 inch high on the prepared baking sheets. Gently brush the gougères with the remaining 2 tablespoons of milk and sprinkle with the remaining 1/4 teaspoon of salt and cup of cheese. Bake for about 8 minutes, until puffed. Rotate the baking sheets from top to bottom and back to front and bake for about 12 minutes longer, until the gougères are golden brown. Serve hot or warm.
The gougères can be baked up to 3 hours in advance. Reheat before serving.