Bring a large pot of water to a boil. Salt it, add the pasta and cook until al dente. Drain, reserving 1 cup of the pasta cooking water.
While the pasta is working, in a small saucepan, add the porcinis and stock (or water). Bring to a simmer. Drain, reserving the stock; coarsely chop mushrooms.
In a large skillet or Dutch oven, heat the EVOO over medium-high heat. Add the butter to melt. Add the fresh mushrooms and cook until browned, 10-12 minutes.
Meanwhile, in another saucepan, heat the milk to barely a simmer. Remove from heat; cover.
Add the carrot, onion, garlic, thyme, sage and bay leaf to the cooked creminis; season with salt and pepper. Cover and cook, stirring, for 5 minutes. Stir in the tomato paste for 1 minute. Stir in the marsala or wine, chopped porcinis, 1 cup of the porcini stock and the warm milk; bring to a simmer.
Add the pasta and reserved pasta water to the sauce. Stir in the cheese and toss to combine.