Wash and dry the oxtails. Add the meat to a large bowl and season with salt and pepper, to taste. Toss the meat in the flour until lightly coated.
Add the olive oil to a large Dutch oven over medium heat. Brown the oxtails on all sides, in batches, until well browned. Transfer the meat to a plate and pour off some of the drippings, if necessary.
Add the onions, carrots, celery, thyme, and garlic to the pot and saute until the onions are tender and beginning to brown, about 5 minutes. Season with salt and pepper, to taste. Stir in the tomato paste and toast. Deglaze the pot with the bottle of red wine and stir up all the browned bits on the bottom of the pan. Stir in the beef broth, tomatoes with their juices and the bay leaves.
Add the browned beef back into the pot along with any accumulated juices that may have collected on the plate. Bring the stew to a boil, then reduce the heat to a low simmer. Cover and cook until the beef is tender and falling off the bone, about 3 1/2 hours.
Stir in the mushrooms and cook uncovered, for 30 minutes more. Taste for seasoning. Add salt and pepper, if needed, and stir in the red wine vinegar to brighten the flavor. Remove the meat from the pot and pull the meat off the bones, then add back to the pot. Discard the bones. Transfer the stew to a serving dish and serve hot over Buttered Noodles.