Preheat the oven to 350 degrees F. Wrap the squash half loosely in aluminum foil and place it on a baking sheet. Bake until tender when pierced with a knife, about 1 hour.
Let the squash stand until just cool enough to handle. Scoop out and discard the seeds. With a large spoon, scoop the pulp from the skin and into a fine sieve. Set the sieve over a bowl to catch the liquid and refrigerate, covering for about 8 hours, or overnight.

Transfer the drained squash to a food processor (a blender will make the squash puree too dense) and process, scraping down the sides of the work bowl once or twice, until smooth. Pass the squash through a fine sieve, using a ladle to force it through.

In a small saucepan, heat 2 tablespoons of the butter over medium heat until foaming. Stir in the cream, salt, nutmeg, and squash puree. Bring to a boil and adjust the level of heat to a simmer. Simmer for 5 minutes. Season with pepper to taste and remove from the heat.

While you prepare the gnocchi, bring the salted water to a boil in a large pot over high heat. Cook and drain the gnocchi. Transfer them to a large warm platter.

In a small saucepan, melt the remaining 2 tablespoons butter over low heat. Pour the melted butter over the cooked gnocchi and sprinkle them with grated cheese. Spoon the gnocchi into shallow bowls, piling them into a little mound. Spoon the squash over the top of the mounds, and let it run down the sides.

Serve immediately.