Place porcini mushrooms in a small shallow bowl with just enough hot water to cover. Let stand 30 minutes.
Place onion in the bowl of a food processor; process until finely chopped. Transfer to a small bowl and set aside. Place carrots and celery in bowl of food processor; process until finely chopped. Transfer to a small bowl (separate from onion) and set aside. Place mixed mushrooms in bowl of food processor; process until finely chopped; set aside.
Heat butter in a large pot over medium-high heat. Add onion and 2 teaspoons salt and 1/2 teaspoon pepper; cook, stirring, until onion begins to brown, about 7 minutes. Add carrots and celery; stir to combine. Cook, stirring occasionally, until mixture is browned, 20 to 25 minutes.
Drain porcini mushrooms and finely chop. Add to pot along with chopped mixed mushrooms. Cook stirring occasionally, until liquid is released and mushrooms are browned, about 10 minutes.
Add tomatoes and cook, stirring occasionally, until liquid is slightly reduced, 6 minutes. Place basil, marjoram, rosemary, and thyme on a piece of cheesecloth, tying with kitchen twine to enclose; add to pot. Bring to a boil and reduce heat to medium-low; simmer for 1 hour.
Add 1 cup water and continue cooking for an additional 1 1/2 hours, stirring occasionally.
Meanwhile, bring a large pot of salted water to a boil. Add pasta and cook according to package directions, until al dente. Drain, reserving some of the cooking water, if desired, and transfer to a large bowl. Add mushroom ragu and toss to combine. Stir in cheese and enough of the pasta water to form a sauce. Garnish with additional cheese and parsley.