Place the dried mushrooms in a small bowl and cover with just enough boiling water to fully submerge the mushrooms. Allow the mushrooms to sit and hydrate for 15 to 20 minutes. Once hydrated, drain the mushrooms (reserving the mushroom broth for later) and finely chop with a knife. Set aside.
Heat ¼ cup of extra virgin olive oil in a large heavy-bottomed saucepan or Dutch-oven over medium heat. Add the diced onion and saute for 8 to 10 minutes, stirring ever few minutes, until the onions are very soft and translucent.
Add the grated carrot and fresh thyme and continue to cook over medium-low heat for an additional 2 to 3 minutes, or until the carrot is very tender. Add the garlic and cook, stirring frequently, for about a minute or until fragrant. Deglaze the pan with the red wine, stirring to scrape up any caramelized bits. Continue to cook until nearly all the red wine has evaporated. Add the crushed tomatoes, chopped hydrated porcini mushrooms, and reserved mushroom broth, stir everything together, and slowly simmer the sauce for 30 to 40 minutes, stirring every so often.
Bring a large pot of salted water to a boil. Boil the pasta until al dente and drain (reserving 1 cup of the starchy pasta water). Toss the pasta with the sauce until it coats the strands well, adding a splash or two of pasta water as necessary.
Serve pasta and top with finely grated parmigiano-reggiano cheese.