Preheat the oven to 500 degrees F.
In a 12 to 14-inch saute pan, heat 2 tablespoons butter over medium-high heat and add the onion, carrot, celery and salt and pepper, to taste, and cook until softened and golden brown. Add the sausage and chicken, and cook until browned. Stir in the tomatoes and cook until they have broken down and released their juices.
In a separate bowl, combine the mushrooms and cheese.
Prepare the soft polenta according to the recipe and pour 1/3 into a buttered casserole. Place a few dots of butter from the remaining 2 tablespoons on top of the polenta, then top it with 1/3 of the meat and tomato mixture, then with 1/3 of the cheese and mushroom mixture. Repeat until all the ingredients are used up. The top layer should be cheese and mushrooms. Cook in the oven for 15 minutes and serve.
In a 3-quart saucepan, heat water and salt until boiling. Drizzle in polenta in a thin stream, whisking constantly, until all polenta is incorporated and mixture begins to thicken. Switch to a wooden spoon, pull off heat and continue to stir until thick as paste.
Stir in the cheese and continue to stir until it is melted and incorporated.
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