Bring 6 quarts of water to a boil in a large pot, and add 2 tablespoons salt.
Meanwhile, in a 10 to 12-inch saute pan, heat the olive oil over medium heat until almost smoking. Add the garlic and saute until lightly browned, 6 to 7 minutes. Remove from the heat and add the Cinzano. Replace the pan on the burner, add the oyster mushrooms and butter, and bring to a boil. Boil until reduced by half. Season with salt and pepper. Remove from the heat and keep warm.
Drop the pasta into the boiling water and cook until tender, 1 to 2 minutes. Drain.
Add the hot pasta to the mushrooms and stir gently over medium heat for 1 minute to coat the noodles. Add the arugula and toss for 30 seconds, or until wilted. Transfer to a warmed serving dish, sprinkle with cheese, and serve immediately.
Mound the flour in the center of a large wooden board. Make a well in the center of the flour and add the eggs. Using a fork, beat the eggs together and then begin to incorporate the flour, starting with the inner rim of the well. As you expand the well, keep pushing the flour up to retain the well shape (do not worry if it looks messy).When half of the flour is incorporated, the dough will begin to come together. Start kneading the dough, using primarily the palms of your hands. Once the dough is a cohesive mass, set the dough aside and scrape up and discard any dried bits of dough.
Lightly flour the board and continue kneading for 10 minutes, dusting the board with additional flour as necessary. The dough should be elastic and a little sticky. Wrap the dough in plastic wrap and allow to rest for 30 minutes at room temperature before using.