Roll the past out to the thinnest setting on a pasta rolling machine and cut the resultant sheet into ribbons 1/4-inch thick. Place on a sheet tray dusted with semolina and cover 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 mushrooms and saute over high heat until browned, about 2 to 3 minutes per batch. The mushrooms should not release their juices. Remove the mushrooms and add the onion, leeks, garlic, thyme leaves and salt and pepper to taste. Cook for 5 more minutes, then add the tomato sauce. Bring to a boil, then reduce to a simmer and add the mushrooms. Cook 10 more minutes and set aside.
Cook the pasta in the boiling water for 1 minute. Drain, reserving the pasta water, and add the pasta to the sauce. Toss over high heat 1 minute to coat, adding a splash of pasta water if necessary to keep the sauce from getting too tight. Divide among six warmed pasta bowls and top with grated Parmigiano-Reggiano.
Basic pasta dough:
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.
Basic tomato sauce:
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic, 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.