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 olive oil over medium heat. Brown ribs, half at a time if necessary, in the hot oil, turning to brown all sides; remove ribs and set aside. In the same Dutch oven cook onions and garlic for 2 minutes, adding another 1 tablespoon of the oil if needed. Add carrots and celery; cook about 5 minutes more or until vegetables are tender.
Return ribs to Dutch oven with the vegetables. Stir in broth, wine, tomato paste, and thyme sprigs. Bring to boiling. Cover; 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 or 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 prepared baking dish. If desired, cut short ribs into smaller portions. Run a thin metal spatula around the edges of the Firm Polenta in the loaf pan. Remove polenta loaf from pan and cut into 1-inch cubes. Arrange cubes evenly over the ribs and sauce in the baking dish. Brush the cubes with the remaining 1 tablespoon olive oil. Bake, uncovered, about 1 hour or until ribs and sauce are heated through and polenta is lightly browned. 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.