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.
Divide the prepared pasta dough into three 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 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.
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds 1 week in the refrigerator or up to 6 months in the freezer. Yield: 4 cups