Preheat oven to 325 degrees F. Sprinkle ribs with salt and pepper. In a 6- to 8-quart oven-going Dutch oven heat 1 tablespoon of the oil over medium-high heat. Cook ribs, half at a time if necessary, in hot oil until brown on all sides. Remove ribs, reserving drippings in Dutch oven. Cook onions and garlic in the reserved drippings over medium heat for 2 minutes, adding another 1 tablespoon of the oil if necessary. Add carrots and celery; cook about 5 minutes more or until vegetables are tender, stirring occasionally.
Return ribs to Dutch oven. Stir in broth, wine, tomato paste, and thyme sprigs. Bring to boiling. Cover Dutch oven; transfer to the oven. Bake for 1 1/2 hours. Stir in mushrooms. Bake, uncovered, for 45 to 60 minutes more or until ribs are tender and sauce is slightly thickened. Cool for 30 minutes. Discard thyme sprigs. Season to taste with additional salt and pepper.
Transfer rib mixture to a large airtight container; cover. Chill for at least 8 hours (up to 3 days).
Preheat oven to 375 degrees F. Lightly grease a 3-quart rectangular baking dish; set aside. Using a spoon, remove any fat from surface of the sauce. Spoon ribs and sauce into the prepared baking dish. If desired, cut ribs into smaller portions.
Using a thin metal spatula, loosen edges of the Firm Polenta from sides of loaf pan. Remove polenta from pan; cut into 1-inch cubes. Arrange cubes evenly over ribs and sauce in baking dish. Brush cubes with the remaining 1 tablespoon oil.
Bake, uncovered, about 1 hour or until ribs and sauce are heated through and polenta is light brown. If desired, sprinkle with parsley.
In a medium saucepan bring the 2 1/2 cups water to boiling. In a medium bowl stir together cornmeal, the 1 cup cold water, and salt. Slowly add cornmeal mixture to the boiling water, stirring constantly. Cook and stir until mixture returns to boiling. Reduce heat to medium-low. Cook 25 to 30 minutes or until mixture is thick, stirring frequently and adjusting heat as needed to maintain a slow boil. Pour into an 8x4-inch loaf pan. Cover; chill at least 8 hours or up to 3 days.