Squeeze the soaked porcinis dry and slice them into 1/4-inch pieces. Strain the soaking water and keep warm. Tie the herb sprigs together with kitchen twine or wrap them in cheesecloth.
In a large skillet, combine the EVOO and 3 tbsp. butter over medium heat. When the butter melts, add the shallots, onion and 1/2 tsp. salt. Cook, stirring often, until the vegetables are softened but not browned, about 7 minutes. Add the porcinis and the fresh mushrooms, the herb bouquet and 1/4 tsp. salt; toss briefly. Increase the heat to medium-high, cover and cook, stirring frequently, until the liquid evaporates and the mushrooms are browned, about 10 minutes. Push the mushrooms to the side of the pan, ad the tomato paste and stir it in the center of the pan for about 1 minute, then mix it into the mushrooms.
When everything is sizzling and just starting to stick, pour in the marsala, stirring constantly as it evaporates. When the mushrooms start sticking to the pan again, pour in the warm mushroom liquid and 2 cups of the hot stock. Bring to a boil, stirring up any browned bits from the bottom of the pan.
Lower the heat to keep the sauce bubbling gently on the surface. Cover and simmer, stirring occasionally, for 20 minutes. Uncover the pan and cook at a high simmer until the mushrooms are thoroughly tender and the liquid is thickened, about 20 minutes. Discard the herb bouquet, scraping off any sauce. Season with salt and pepper.
Meanwhile, in a large dutch oven or other heavy pot, bring 5 cups water, the bay leaves, remaining 1 tbsp. butter and 2 tbsp. coarse salt to a simmer over medium heat. Gradually sift in the polenta through the fingers of one hand, stirring constantly with a wooden spoon or whisk. Lower the heat to medium-low and stir until the polenta is smooth and thick and pulls away from the sides of the pan, about 20 minutes. (If using instant polenta, cook according to package directions.) Discard the bay leaves. Let the polenta rest for 5 minutes. Using a wet spoon, portion the polenta into dishes. Top with a spoonful of the mushroom ragu and some cheese.