Begin by cooking the pasta in a large pot of boiling salted water. Make sure your pot is large enough to allow the pasta to cook without sticking together. Cook it for about 12 minutes, stirring frequently. Once the pasta is cooked, drain it through a colander and toss it with 1 tablespoon of olive oil. This prevents the pasta from clumping together as it cools.
In a large sauté pan, heat 2 tablespoons oil over medium heat. In a food processor, pulse onions, celery, carrots, garlic, and walnuts a few times until they are well minced. Transfer to a pot and sauté for about 3-4 minutes, stirring occasionally. Wipe down the food processor and add the mushrooms. Pulse a few times until mushrooms are broken down and resembled minced meat. Add them to the veggies and sauté for another 3-4 minutes until the mushrooms have cooked down in size.
Add the wine and bring to a simmer. Simmer for 5-7 minutes until the wine has cooked all the way down. Stir in the non-dairy milk and continue to cook it down for another 5-6 minutes until the milk has evaporated.
Once the milk has cooked down, add the tomatoes, tomato paste, fresh herbs, salt, and pepper. Cook, uncovered, until the sauce is rich and thick. Remove from heat. Taste and adjust seasonings before serving.