Grease the walls of a Crock Pot with 1 tablespoon of the butter. Measure remaining ingredients; add to slow cooker with remaining butter. Stir well; cover and cook on low for 3 to 4 hours, stirring occasionally.
Grease a 13x9 baking pan. Pour half the polenta into the pan and spread evenly. Grease both sides of a piece of parchment paper, lay on top of the polenta layer, then spread remaining polenta on top. Refrigerate at least 2 hours to set.
In a Dutch oven, brown the sausage, then drain and reserve. Cook the chopped vegetables in the olive oil until softened. Add the sausage, the diced tomatoes, and the italian seasoning, and simmer for 30 minutes, stirring occasionally. Add the mushrooms, tomato paste, and basil, and simmer for 10 more minutes.
To assemble, lift off top layer of polenta. Add half the sauce, then top with removed polenta. Add the second half of the sauce, then top with cheese.
Bake in 350F oven until cheese is melted and lasagna is warmed through - 15-20 minutes if sauce was hot, 30-45 if refrigerated.