Mushroom Lovers Polenta-Topped Shepherd's Pie

Place dried porcini in a small pot and cover with stock. Bring to a low boil, reduce heat and gently simmer to reconstitute.
Meanwhile, heat oil, 4 turns of the pan, over medium-high heat. Add diced mushrooms and brown 12-15 minutes, stirring frequently to develop their beefy flavor.  Add carrots, celery, onion, garlic, thyme, bay salt and pepper, and stir. Cook partially covered, stirring occasionally, to soften, 10-12 minutes more.  Stir in tomato paste, cook 1-2 minutes add Marsala or wine and let evaporate, about 1 minute. Add reconstituted porcini and all but last few spoonfuls of stock to prevent settled grit being added.  Stir in milk and simmer gently over low heat while you prepare the polenta.

Heat the stock and milk in saucepot for the polenta. Whisk in the polenta and cook 2-3 minutes, until creamy and smooth, adding more stock or milk if it gets too stiff.  Stir in butter and season with salt, pepper and nutmeg. Add half of the grated cheese and remove from heat.

Preheat broiler with rack at the center of the oven.

Transfer the mushroom filling to baking dish, individual bowls or large ramekins and top with polenta. Sprinkle with more Parm and brown under broiler, serve.