Preheat the oven to 400 degrees F.
Cover the dried mushrooms with about 2 cups water in a small pot. Bring to boil, then reduce the heat and simmer to reconstitute them.
Meanwhile, heat the EVOO in a Dutch oven over medium-high heat. Add the portobello and mixed fresh mushrooms and cook until darkened and tender, 15 to 20 minutes. Add the garlic, rosemary, thyme, carrots and onions, and season with salt and pepper. Cook, stirring, 7 to 8 minutes. Add the tomato paste and stir 1 minute. Add the Marsala and deglaze the pan, scraping up the browned bits from the bottom. Add the soaked dried mushrooms and most of the cooking broth, leaving behind the last few spoonfuls as grit may settle there. Transfer the stew to a 10 to 12-inch casserole dish.
Heat the milk and 1 1/2 cups water to a simmer in a large saucepan. Whisk in the polenta and cook, whisking, until thickened. Remove from the heat while the polenta is still pourable and stir in the butter, Parmesan and salt and pepper to taste.
Pour the polenta over the mushrooms in the casserole dish. Top with the smoked cheese and sprinkle with parsley.
Cover the casserole with foil and bake for 20 minutes. Uncover and bake until golden and bubbly on top, 15 to 20 more minutes.
If you are not serving the casserole immediately, assemble it but do not bake. Cool, then store in the refrigerator. Bake as directed before serving.