Place the dried porcini in a small pot and cover with stock. Bring to a low boil, then reduce the heat and gently simmer to reconstitute.
Meanwhile, heat the oil, four turns of the pan, over medium-high heat. Add the diced mushrooms and brown for 12-15 minutes, stirring frequently to develop their beefy flavor. Add the carrots, celery, onion, garlic, thyme, bay leaf, salt and pepper and stir. Cook, partially covered, stirring occasionally, to soften, 10-12 minutes more. Stir in the tomato paste; cook for 1-2 minutes, then add the Marsala or wine and let evaporate, about 1 minute. Add the reconstituted porcini and all but last few spoonfuls of stock to prevent settled grit from being added. Stir in the milk and simmer gently over low heat while you prepare the polenta.
Heat the stock and milk in a saucepot for the polenta. Whisk in the polenta and cook for 2-3 minutes, until creamy and smooth, adding more stock or milk if it gets too stiff. Stir in the butter and season with salt, pepper and nutmeg. Add half of the grated cheese and remove from the heat.
Pre-heat the broiler with a rack at the center of the oven.
Transfer the mushroom filling to a baking dish, individual bowls or large ramekins and top with the polenta. Sprinkle with more Parmigiano Reggiano and brown under the broiler; serve.