Preheat the oven to 400 degrees F.
Core the apples, maintaining the peel, and place on a sheet pan. Roast in the oven 20 minutes, until soft and juicy. Remove and allow to cool. Squeeze flesh from peel and discard peel.
In a large bowl, combine the apple flesh with the ricotta, egg, amaretti, 1/4 cup cheese and salt and mix well to combine.
Roll out the pasta to the thinnest setting on a pasta rolling machine, then cut into 2-inch squares. In center of half of the squares, place a scant tablespoon full of the apple and cheese mixture. Moisten the edges of these squares with water, then top each square with an un-filled square, and press with your fingers to seal the edges.
Bring 6 quarts of water to a boil and add 2 tablespoons salt.
In a 12-to 14-inch saute pan, heat the butter until it foams and subsides.
Meanwhile, drop the pasta in the boiling water and cook 3 minutes. Drain the pasta, reserving the cooking liquid, and add the pasta to the pan with the butter. Add a splash of the pasta water if necessary to keep the sauce from becoming too "tight." Toss over high heat 1 minute to coat. Divide pasta evenly among 6 warmed pasta bowls, top with grated Parmigiano-Reggiano, 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 reflour the board and continue kneading for 6 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.