Rehydrate the dried mushrooms in hot water for at least 20 minutes; strain and set aside.
Melt oil and 1 Tbsp butter in a skillet over medium heat. Sauté the shallots (or onion) until they translucent but not brown. Add the rehydrated, strained mushrooms to the shallots and sauté for 1 minute so that flavors begin to combine. Set aside.
In a saucepan, slowly heat milk to just a bubble. Slowly whisk in the polenta. Add the Ammazza Suocera. Cook polenta for 20-25 minutes.
Add the shallot-mushroom mix to the polenta. Season with more Ammazza Suocera to taste. Transfer the mixture into a buttered loaf pan and cool in refrigerate until firm.
Meanwhile, preheat oven to 350°F. Cut chilled polenta into 1-inch thick slices and arrange in a non-stick baking dish. Melt the remaining butter and drizzle it on top of the polenta slices. Sprinkle cheese on top.
Transfer baking dish to oven and bake for 15-20 minutes.