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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.
Bring 6 quarts of water to a boil and add 2 tablespoons salt.
In a 12 to 14-inch saute pan, heat the olive oil over high heat until smoking. Add the onions and cook over high heat for 5 minutes, until onions are light golden brown. Add the mushroom pieces and continue cooking until they have given up most of their water, about 9 to10 more minutes. Add the wine and the tomato sauce, butter and salt and pepper to taste and reduce heat to medium. Cook for 5 more minutes, until mixture is of a ragu consistency.
Meanwhile, cook the pasta in the boiling water for 1 minute. Drain the pasta, reserving the cooking water, and add the pasta to the ragu. Toss over high heat one minute to coat the pasta. Divide the pasta evenly among four warmed pasta bowls, grate Parmigiano over and serve immediately.