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Dry the ribs thoroughly with paper towels and season generously with salt and pepper. Add 2 Tbsp. olive oil to an oven-safe heavy pot or dutch oven and heat over high heat. Once the oil is shimmering, add the short ribs and brown on all sides, approximately 2-3 minutes per side. The pot should not be overcrowded. Work in batches if necessary.
While the short ribs are browning, preheat the oven to 375 degrees.
Puree all vegetables and garlic in a food processor or blender until finely chopped into a thick paste.
Once ribs are browned, remove the ribs to a plate and tent with foil to keep warm. Drain the fat from the pot and add remaining 2 Tbsp. olive oil.
Add the pureed vegetables and season generously with salt. Brown over high heat, without stirring, for 5 to 7 minutes.
Add the tomato paste, stir, and let the vegetables brown for an additional 3 to 5 minutes. Add the wine and, using a wooden spoon, scrape any bits from the bottom of the pan. Stir occasionally, letting this mixture reduce by half.
Once reduced, return the short ribs and any accumulated juices to the pot along with enough water to almost cover the meat. Add the thyme and bay leaves. Cover the pot and place it in the hot oven. Allow to cook for 2 1/2 hours, checking periodically and adding more water if needed. Remove the lid and cook for 30 minutes longer, to allow the sauce to further reduce.
Transfer the ribs to a serving platter or dish. Let the sauce and solids sit in the pot for a few minutes to cool and with a shallow spoon, skim off as much of the fat as possible from the surface. Season the sauce to taste with salt and pepper, and add the vinegar.
Serve the ribs on top of creamy polenta with additional braising liquid.