1. Arrange one oven rack in the top third of the oven and the other in the bottom third. Preheat the oven to 400 degrees F. Line 2 large baking sheets with parchment paper.
2. In a large saucepan, heat the milk, water, butter, salt, and cayenne over medium heat to a simmer, stirring to melt the butter. Turn off the heat and add the flour all at once. Stir vigorously until smooth and a thin film of dough sticks to the bottom of the pan. Transfer the dough to a mixing bowl.
3. Using an electric mixer, beat the dough on low speed for 1 minute to cool it down. Raise the speed to medium and add the eggs, one at a time, beating very well after each addition. Continue beating, scraping down the bowl occasionally, until the batter is shiny, smooth, and elastic. Reduce the speed to low and beat in the cheese and chives.
4. Drop the batter by teaspoons onto the prepared baking sheets, spacing the mounds 1 inch apart. Bake for 15 minutes, switch the positions of the baking sheets, and bake for 12 to 15 minutes longer or until golden brown, domed, and firm to the touch. Serve warm.