Preheat the oven to 350 °F
Cut the squash in half, remove the seeds, drizzle with the olive oil, and place on a baking sheet. Roast for 25-35, or until the squash is very soft. Remove from the oven, let cool, the scoop the flesh from the skin.
In a large bowl, combine the cooled squash, cheese, nutmeg, balsamic vinegar, and salt and pepper. Stir well to combine.
Roll out the pasta dough to the thinnest setting on a pasta machine. Using a biscuit cutter or water glass, cut out 2-inch circles. Pipe or carefully spoon a rounded tablespoon of filling onto the center of half of the rounds and cover the filling with a second pasta round. Press the edges together firmly to seal.
Bring 6 quarts of water to a boil and add 2 tablespoons of salt. Drop the lune in the boiling water and cook for 2 minutes. While the pasta cooks, melt the butter in a 12- to 14- inch sauté pan until it foams and subsides. Add 2 to 4 tablespoons of the pasta cooking water and whisk to emulsify. Drain the pasta and add it to the butter. Add the sage leaves and toss gently for 1 minute over medium heat to coat the pasta with sauce. Divide the lune among four warmed plates, grate the Parmigiano-Reggiano and amaretti over each plate, and serve immediately.