Position racks in the top and middle of the oven and heat the oven to 425ºF. Line two baking sheets with parchment.
In a medium heavy-based saucepan, heat the water, butter, salt, cayenne, and nutmeg over high, stirring to melt the butter. Bring to a boil and then dump in all the flour at once.
Take the pan off the heat and stir vigorously with a wooden spoon until you get a smooth, thick paste. Put the pan back on the stove, reduce the heat to low, and stir another minute or so to cook off more moisture. The dough should start to form a shiny ball and pull away from the sides and bottom of the pan.
When the dough is dry enough, take the pan off the heat. (Transfer to a stand mixer bowl now, if using.) Pour in 1 egg and then beat until it's well blended and the dough is smooth again. Repeat five more times and then start to test the dough's consistency: it should fall from the spoon in a graceful "plop." If it seems too stiff, whisk up the last egg and add a bit of it and test again. (Note: if you're using a stand mixer, use the paddle attachment. Use only low speed and don't overmix or the puffs will be tough.)
Add the pecans and Cheddar to the dough and carefully fold to distribute them. With a mini ice-cream scoop or two tablespoons, drop mounds about the size of a whole walnut shell onto the baking sheets, spaced about 1 inch apart. Sprinkle the shredded Parmigiano on top.
Bake in the heated oven until puffed, deep golden brown, and just barely moist inside, 25 to 30 minutes (you'll have to break one open to really check the doneness). Switch the positions of the baking sheets after 15 minutes for even baking. Transfer the gougères to a cooling rack. Repeat with any remaining dough. Serve when just barely warm or at room temperature.
You can make the dough a day ahead, keep it covered in the refrigerator, and then scoop and bake not more than an hour before serving so they’re really fresh. They’re still delicious when baked further ahead than that, but they do tend to soften a bit.