Divide the prepared pasta dough into 3 balls of equal size. Roll each ball out to the thinnest setting on a pasta rolling machine. Then, using the machine or a sharp knife, cut each sheet of pasta into ribbons of 1/3-inch thickness. Place the ribbons on a sheet tray dusted with flour, and cover the tray with a clean, damp dish towel until ready to cook.
Bring 6 quarts water to a rolling boil and add 2 tablespoons salt.
In a 12 to 14-inch saucepan, heat the olive oil over a medium-high flame until hot but not smoking. Add the onion and sweat over medium heat until soft and translucent, about 5 minutes.
Meanwhile, add the chicken livers to the pan and continue cooking until the livers are slightly browned on all sides, about 5 minutes more.
Add the wine and continue cooking over medium-high heat until the wine evaporates. Continue cooking over medium heat for 10 minutes more.
Add the pasta to the boiling water and cook until soft but al dente, about 5 to 6 minutes. Drain the pasta, reserving 1/2 cup of the cooking water, and add the pasta to the pan with the chicken livers. Add parsley and toss over high heat to combine. Season with salt and pepper. Serve the pasta in 4 warmed pasta bowls, each topped with generous truffle shavings, and grated Parmigiano, to taste.
Sift together and then mound 2 1/4 cups of the flour and a pinch of salt in the center of a large wooden cutting board. Make a well in the middle of the flour and add the egg yolks. Using a fork, beat together the egg yolks 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. Knead for about 5 to 6 minutes, adding any of the remaining flour if necessary to create a cohesive mass. 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.