Place the dried porcini in a heatproof bowl and cover with the boiling water. Let soak until softened, 20β30 minutes, then lift out, finely chop, and strain the soaking liquid through a fine sieve or coffee filter; reserve both.
Heat the olive oil and butter in a wide heavy pot over medium heat. Add the onion, carrot, and celery with 0.5 tsp of the kosher salt and cook, stirring, until very soft and lightly golden, 8β10 minutes.
Stir in the garlic, fresh thyme leaves, and fresh rosemary and cook until fragrant, about 1 minute.
Add the cremini mushrooms and shiitake mushrooms, season with another 0.5 tsp kosher salt, and cook over medium-high heat, stirring occasionally, until they release their liquid and the pan starts to brown, 10β15 minutes.
Stir in the chopped porcini mushrooms, then add the tomato paste and cook, stirring, until the paste darkens and coats the mushrooms, 2β3 minutes.
Pour in the dry red wine and simmer, scraping up browned bits, until nearly evaporated, 3β5 minutes.
Add the canned crushed tomatoes, the reserved porcini soaking liquid (leave any grit behind), the vegetable stock, bay leaf, black pepper, and the remaining 0.5 tsp kosher salt. Bring to a gentle simmer, partially cover, and cook, stirring occasionally, until thick, glossy, and deeply flavored, 25β35 minutes. Sauce is done when it clings to a spoon and mushrooms are tender.
Remove and discard the bay leaf. Stir in the flat-leaf parsley and let the ragu rest off heat for 5 minutes; adjust salt and pepper to taste.
Serve the mushroom ragu hot over pasta or polenta and finish with Parmigiano Reggiano, finely grated (for serving).