Finely chop the onion, celery, garlic, and carrot. In a large pot, heat a few tablespoons olive oil over medium heat.Cook the chopped vegetables, stirring often, about 5 minutes. In a food processor, finely mince the crimini mushrooms. Add them to the pot with the vegetables and cook, stirring, until the excess water has been cooked out, about 10 minutes.
Meanwhile, in a saute pan, toast the walnuts over medium heat until just browned and fragrant, stirring often, about 7 minutes.
Using the food processor, pulse the toasted walnuts until minced but not a powder, about 6-7 pulses.Stir the minced walnuts into the vegetables and mushrooms. Stir the white wine, tomato paste, bay leaf, and nutmeg into the vegetables and cook over medium-low heat until the wine is almost totally evaporated.
Stir in the vegetable or chicken stock, milk, and 1 teaspoon kosher salt. Cook over medium-low heat until the liquid is reduced by half, about 30 minutes.
Bring a large pot of water to a boil. Cook your pasta for about 2 minutes less than package instructions, then transfer to the pot with the sauce.Reserve 1 cup of pasta water in case the sauce is too thick. Stir the pasta into the sauce, along with the grated parmesan, and cook until the pasta is cooked and well combined with the sauce, about 2 minutes. If the sauce is too thick to evenly coat the pasta, stir in some of the reserved pasta water.
Serve the pasta, topped with extra Parmesan cheese and a scoop of ricotta.