Preheat the oven to 350 degrees F.
Cut the pumpkin in half, remove the seeds, drizzle with olive oil, and place on a baking sheet. Roast for 25 to 35 minutes or until the squash is very soft. Remove from the oven, let cool, then scoop out the flesh from the skin.
In a large bowl, combine the cooled pumpkin, 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 scant teaspoon of filling onto the center of half of the rounds and cover the filling with the 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 saute pan until it foams and subsides. Add 2 to 4 tablespoons of the pasta cooking water to the butter 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 4 warmed plates, grate the Parmigiano-Reggiano and amaretti over each plate, and serve immediately.
Mound 3 1/2 cups of the flour in the center of a large wooden cutting board. Make a well in the middle of the flour and add the eggs and the olive oil. Using a fork, beat together the eggs and oil and begin to incorporate the flour, starting with the inner rim of the well.
As you expand the well, keep pushing the flour up from the base of the mound to retain the well shape. The dough will come together when half of the flour is incorporated.
Start kneading the dough with both hands, using the palms of your hands. Once you have a cohesive mass, remove the dough from the board and scrape up and discard any leftover bits. Lightly re-flour the board and continue kneading for six more minutes. The dough should be elastic and a little sticky. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature. Roll or shape as desired.
Preheat the oven to 300 degrees F.
Place the almonds, cornstarch, and confectioners' sugar in the bowl of a food processor and pulse until the almonds are very finely chopped. Place the egg whites in the bowl of an electric mixer and add the salt. Using the whip attachment on medium speed, beat the whites until foamy and light. Gradually add the granulated sugar in a steady stream, continuing to beat until the egg whites are somewhat stiff and glossy. Beat in the almond extract and amaretto.
Pipe or drop about 2 tablespoons batter (how much) onto an ungreased cookie sheet about 2 inches apart. Sprinkle a little turbinado sugar on top of each cookie. Bake for 15 minutes or until the cookies have begun to turn pale golden and are just beginning to crack slightly. Lower the oven temperature to 200 degrees F, and leave the oven door ajar to release some of the heat. Leave the cookies in the oven for at least 25 to 30 minutes, or until they are completely dry and crisp in the center. Allow the cookies to cool briefly on the baking sheets, then transfer them to a rack to cool completely. Store in an airtight container.
Yield: about 30 cookies