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, cover the porcinis with the stock (or water). Bring to a simmer. Drain, reserving the stock; coarsely chop.
In a large skillet or dutch oven, heat the EVOO over medium-high heat. Add the butter to melt. Add the creminis and cook until browned, 10 to 12 minutes.
Meanwhile, in another saucepan, heat the milk to barely a simmer. Remove from the 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, porcinis, 1 cup of the porcini stock and the milk; simmer.
Add the pasta and reserved pasta water to the sauce. Stir in the cheese.