Pre-heat 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 minutes, or until the squash is very soft. Remove from the oven and let cool, then 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 using 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 six quarts of water to a boil and add two tablespoons of salt. Drop the lune in the boiling water and cook for two minutes. While the pasta cooks, melt the butter in a 12-14-inch sauté pan until it foams and subsides. Add 2-4 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 one 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.
*Basic Pasta: Make a mound of the flour. Create a well in the middle, add the eggs and mix with a fork, slowly incorporating the flour. (Alternatively, mix all together in a Kitchen Aid mixer).If too dry, add a fourth egg yolk. If still too dry, add the white. Once it forms a ball, knead for 15-20 minutes. Wrap in plastic and rest for at least 30 minutes. You can now cut off a portion (about a quarter) and roll out on a pasta machine.