Preheat oven to 375 degrees F.
On a sheet pan, place the squash flesh side down and roast until very tender (about 45 minutes). At the same time, bake potatoes directly on the rack of oven for 1 hour.
Split the potatoes and allow to cool slightly, or until you can handle them. Don't let them cool completely. Scoop the flesh of the potatoes and the squash into a bowl and mash with a hand masher. Mix in the egg, salt and nutmeg. Then add the flour and mix until a soft dough forms. Do not do this in a mixer, it will overwork the dough. Add flour by the spoonful if it's still too moist.
Turn out onto a floured board and divide into 8 portions. Roll out into ropes and cut into 1/2-inch pieces. Line the pieces up on a floured sheet pan as you work. At this point you could freeze them on the pan until solid, then transfer to zip top bags and store in the freezer.
In a large pot of boiling, salted water gently drop in the dumplings. Don't overcrowd. As they begin to float, remove them with a slotted spoon and toss them into an ice bath.
Drain off the water and toss in a little oil. Store loosely in containers until ready to use.
To reheat, in a saute pan over high heat add 1 tablespoon of soft butter. Cook until the butter begins to foam and turn brown. Add 2 teaspoons sage leaves and 1 cup of dumplings. Cook for an additional minute until the dumplings are heated through. Repeat until you have desired amount of servings. Plate and top with freshly grated Parmesan.
Recipe courtesy of Alton Brown