In a small, heavy-bottomed skillet, add the walnuts over medium heat. Cook, stirring occasionally, until lightly toasted, 2–4 minutes. Remove and let cool slightly, then finely chop.

Position racks in the top and bottom thirds of the oven and preheat to 425°. Line two baking sheets with parchment paper or silicone baking mats.

In a medium saucepan, add the water, milk, butter, and salt; bring to a rapid boil. Add the flour all at once, lower the heat, and stir rapidly with a wooden spoon or whisk until a dough forms. Keep stirring to dry the dough slightly, 1–2 minutes more.

Transfer to the bowl of a stand mixer fitted with the paddle attachment, or a medium bowl. Let sit for a minute, then add the eggs one by one, beating until incorporated. (Dough may appear to separate, but it will come together by the time the last egg goes in.) Beat in the mustard, then the cheeses and walnuts.

Using a small (1 ¹⁄₂ teaspoon) cookie scoop or spoon, scoop the dough onto the prepared baking sheets, leaving 2 inches between each. Transfer to the oven and immediately turn down the temperature to 375°. Bake for 12 minutes, then rotate the pans from front to back and top to bottom. Continue baking until the gougères are puffed, golden, and firm enough to pick up, 15–20 minutes more. Remove and serve immediately.