- Brown ribs in oil in a stockpot on all sides. Remove to a paper towel-lined plate.
- Remove pan from heat and carefully add sherry and sherry vinegar. Return to heat and cook, scraping up any browned bits from pan’s bottom, for about 5 minutes.
- Add remaining ingredients, except butter, to the pot. Return browned ribs to the pot. Bring liquid to a simmer, cover, and cook for about 3 hours. Season with salt and pepper.
- Remove ribs from pot and set aside. Strain liquid and return to pot. Add the butter and stir until incorporated. Add ribs back to the pot, coating well with sauce.
- For polenta cake, bring water, milk, and salt to boil. Whisk in polenta and cook 25−30 minutes until very thick, stirring frequently. Remove from heat and stir in 1 tablespoon butter and parmesan. Grease a 9×9-inch square baking dish and spread polenta evenly into dish. Cover with plastic and refrigerate until dinner. Cut it in rounds and sear both sides in remaining butter when ready to serve.
- For ragout, heat oil in a large frying pan over medium heat. Add shallots and garlic and cook, stirring constantly, for 3−4 minutes or until the shallots are soft. Add mushrooms and cook, stirring frequently, for 2−3 minutes or until mushrooms are tender. Stir in tomatoes. Reduce the heat to medium low. Simmer, stirring occasionally, for 2 minutes. Remove from heat and stir in the chives. Season.
- To serve, set ribs atop polenta cake and top with ragout.