1. Heat the oven to 400 degrees Fahrenheit, with a rack in the middle. Place the potato directly on the middle rack. Toss the squash with oil, 1/2 teaspoon salt, and a few grinds of pepper. Spread in a single layer on a baking sheet and place on the same rack.
2. Bake the squash, stirring halfway through, until it’s tender but not brown. Turn the potato and continue baking for 30 to 40 minutes longer until soft when squeezed. When the squash is cool, mash in a wide bowl with a large fork or potato masher.
3. When the potato is done, let it cool and peel off the skin. Add the potato to the bowl with the squash, and mash, using whatever you used to mash the squash. Mix until you no longer see any spots of potato.
4. The mixture should be cool by now, but if not, let it cool a bit more. Use the fork to incorporate the cheese, then make a well in the center. Add the eggs, 1/2 teaspoon salt, nutmeg, and cayenne. Beat with the fork, and gradually incorporate the squash mixture, continuing until it is thoroughly mixed, switching to a rubber spatula.
5. Sprinkle the flour over the top of the mixture and mix well. Briefly knead in the bowl with floured hands to form a smooth dough, and turn out on a lightly floured work surface. If the dough feels very sticky, add more flour, one tablespoon at a time, until it’s no longer sticky.
6. Line a baking sheet with wax paper and flour the paper. Divide the dough into four pieces. Roll each piece into a rope about 3/4 inch in diameter. If it becomes too long to manage, cut the rope in half. Cut the rope into one-inch pieces and transfer to the baking sheet. The gnocchi can be refrigerated for a few hours or frozen for up to three months (freeze the gnocchi on the baking sheet and then transfer them to a zip-top bag).
7. Bring a large pot of water to a boil and add a generous amount of salt. Add half the gnocchi. They will float to the top after about a minute but let them cook for two to three minutes longer. Use a slotted spoon to transfer them to a paper towel–lined plate. Remove the towel after a minute so they don’t stick. Repeat with the remaining gnocchi.