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Divide the prepared pasta dough into 3 equal-sized balls and roll each ball out to the thinnest setting on a pasta rolling machine. Cut each sheet of pasta into 4-inch squares. Place a tablespoon of the chestnut filling in the center of each square and fold the squares, corner to corner, then join the remaining corners to form the tortelli. Set the tortelli on a sheet tray dusted with flour, and cover with clean, damp kitchen towels.
Bring 6 quarts of water to a boil and add 2 tablespoons salt. Drop the tortelli into the boiling water and cook 2 minutes.
Meanwhile, in a 14 to 16-inch saute pan, heat the butter until it foams and subsides. Add the sage and allow to rest. Drain the pasta, reserving the cooking liquid, and add to the saute pan with the butter. Add a few splashes of the pasta cooking water and toss over high heat 2 minutes to coat and emulsify the sauce. Divide the pasta evenly among 4 warmed pasta bowls, top with grated Parmigiano-Reggiano and serve immediately.
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 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.