1. Mince the vegetables and cook: In a food processor, combine the onion, carrot, and celery and pulse until minced. In a large Dutch oven or pot, heat 3 tablespoons of the olive oil over medium heat. Transfer the minced vegetables to the pot. Add the mushrooms to the food processor, pulse until minced, then add them to the pot. Cook, stirring occasionally, until the vegetables are very soft, 8 to 10 minutes.
2. Build the rest of the sauce: Add the remaining 1 tablespoon olive oil, garlic, and tomato paste to the pot and cook, stirring continuously, until the tomato paste darkens slightly, 2 to 3 minutes. (Reduce the heat, if needed, to prevent the garlic from burning.) Pour in the red wine and let boil for 1 minute, stirring to scrape up the brown bits from the bottom of the pot. Add the tomatoes, tamari, oregano, salt, and pepper. Stir to mix. Reduce the heat to low, cover with a lid, and simmer for at least 15 minutes for the flavours to meld.
3. Meanwhile, cook the pasta: Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until al dente according to the package directions. Scoop out and reserve 1 cup of pasta water. Drain the pasta.
4. Combine: Add the pasta to the sauce and toss to coat. Sprinkle in the Parmesan. Toss, adding more reserved pasta water if needed, a little at a time, until the noodles are coated in a silky sauce. Divide the pasta among bowls and garnish with more Parmesan.