Heat oven to 375ºF.
Bake the potatoes directly on the middle rack of the oven until very tender, about 1 hour. At the same time, place the squash, flesh-side down, on a half-sheet pan and roast until very tender, about 45 minutes.
Split the potatoes and allow to cool slightly, or until they can be handled safely. Don't let them cool completely. Scoop the flesh of the potatoes into a large mixing bowl and smash with a potato masher.
Scoop the squash from its skin and add to the potatoes. You should have approximately 10 ounces of potato and 20 ounces of squash. Mash them together with a hand masher until mostly smooth. There may be some small lumps. Mix in the egg, salt, and nutmeg. Then add the flour and mix with a large wooden spoon until a soft dough forms. Do not do this in a mixer, it will overwork the dough. Add more flour by the spoonful if the dough is still too moist.
Turn dough out onto a clean, floured counter and divide into 8 portions. Roll each portion out into ropes, about 2 feet long, and cut into 1-inch pieces. Line the pieces up on a floured half-sheet pan as you work. At this point, you could freeze them on a half-sheet pan until solid, then transfer to zip-top bags and store in the freezer for up to 1 month.
In a large pot of boiling, salted water, gently drop in the dumplings about 2 dozen at a time; don't overcrowd. As they begin to float, 3 to 4 minutes, remove them with a slotted spoon and toss them into an ice bath for 1 minute.
Drain the dumplings, pat dry with a clean kitchen towel, and toss in a little oil. Store loosely in a large container until ready to use. The dumplings can be cooked and stored in the fridge this way for up to 3 days.
To serve: Melt 1 tablespoon of butter in a large sauté pan over high heat. Cook until the butter begins to foam and turn brown. Add 2 teaspoons sage leaves and 1 cup of dumplings. Cook until the dumplings are heated through, about 1 minute. Repeat until you have desired amount of servings. Plate and top with freshly grated Parmesan.