Preheat the oven to 375°F (190°C). Line 2 baking sheets with parchment paper.
Put the water, butter, and salt in a medium, heavy saucepan and place over medium-high heat. Cook, swirling the pan a few times until the butter melts; once it has melted, increase the heat to high and bring the mixture to a rolling boil. Turn off the heat and dump in the flour. Beat the mixture with a wooden spoon until it starts to pull away from the sides of the pan.
Take the pan off the stove, set it on a counter or hot pad, and let it cool, without stirring, for 5 minutes.
Now, work in the eggs, 1 at a time, beating the mixture well with the spoon after each addition (you have to put some muscle into it to incorporate the eggs fully). With each addition, the dough should look glossy and slick at first, then stick to the sides of the pan before you add the next egg. After beating in the last egg, beat in the dry mustard and cayenne, then the cheese and chives.
Scoop up a heaping teaspoon of dough and with another spoon, push it off onto the paper-lined baking sheet (it should form a mound about 1 inch in diameter). Continue with the remaining dough, leaving an inch of space between the gougeres (work in batches as necessary, the dough can stand, covered with buttered waxed paper or parchment, for up to 1/2 hour).
Bake until the gougeres are puffy and light golden, about 25 minutes, switching pan positions halfway through. Remove from oven and let cool slightly before serving, or turn off oven and let gougeres remain in oven, with door ajar, for up to 1 hour.