Starting the night before, preheat the oven to 350 degrees. Halve the squash, scoop out the seeds, wrap it loosely in foil, place on a baking sheet, and bake until tender when pierced, about 35 minutes. Scoop out and discard the seeds. With a large spoon, scoop the pulp from the skin directly into a fine sieve. Set the sieve over a bowl to catch the liquids, cover and allow the squash to drain overnight in the refrigerator.

Next day, boil potatoes, peel and rice them while still hot. Salt them lightly and let completely cool on a working surface.Meanwhile, puree the squash. Add eggs, salt, nutmeg, cheese and squash puree. Add some flour, mix well and begin kneading. Work the dough quickly, until all the flour is incorporated and the dough is homogenous. Dust the dough, your hands, and the work surface lightly with flour and cut the dough into four equal parts and set on the side. Work one piece of dough, continue to dust dough, hands, and surface as long as the dough feels sticky.

Using both hands, roll the piece of dough into a rope ½-inch thick, then slice the ropes at ½-inch intervals.  Sprinkle some flour and roll each piece into a ball, flouring as needed. Holding a fork at a 45-degree angle to the table, indent each dumpling with your thumb on the tynes of the fork to produce a ribbed effect and an indentation where your thumb was. Set on a floured sheet pan, and continue with all of the dough until finished.

To cook gnocchi, bring 6 quarts of water with 2 tablespoons of the sea salt to a vigorous boil. Drop the gnocchi into boiling water a few at a time, stirring gently and continuously with a wooden spoon, and cook for 2-3 minutes, until they rise to the surface.

Remove the gnocchi from the water with a slotted spoon or skimmer, transfer them to a warm platter, adding a little sauce of choice, and boil the remaining pieces in batches until they are all done.