Braise short ribs: Put oven rack in lower third of oven and preheat oven to 250°F.
Pat beef dry. Heat oil in a wide (12 inches in diameter) 3- to 5-quart heavy pot over moderately high heat until hot but not smoking, then brown beef on all sides, turning with tongs, about 8 minutes. Transfer to a plate and sprinkle with 1/4 teaspoon sea salt and 1/4 teaspoon pepper.
Add chopped carrots, onion, and garlic to oil in pot and cook over moderate heat, uncovered, stirring occasionally, until softened, about 5 minutes. Stir in 1 cup puréed tomatoes (reserve remainder for another use) and bring to a boil over moderately high heat. Add wine and boil, stirring occasionally, until sauce is thickened, about 8 minutes.
Add veal stock, thyme, bay leaf, vinegars, and remaining 1/2 teaspoon sea salt and 1/4 teaspoon pepper to sauce, and bring to a simmer. Skim fat from surface, then add beef along with any juices accumulated on plate and cover pot with a tight-fitting lid. Transfer to oven and braise until beef is very tender, 4 to 5 hours.
Cook vegetables while beef braises: Blanch pearl onions in a wide 2- to 3-quart heavy saucepan of boiling water 1 minute, then drain in a sieve. When just cool enough to handle, peel onions with a paring knife, trimming root end just enough to leave onions intact.
Heat butter in dried saucepan over moderate heat until foam subsides, then cook onions, stirring occasionally, until brown spots appear, 3 to 5 minutes. Stir in vinegar, then add chicken stock and carrots and bring to a boil. Reduce heat and simmer, covered, until vegetables are tender, 20 to 25 minutes. Uncover and boil, if necessary, until liquid glazes vegetables.
While vegetables are simmering, cook bacon in a 10-inch heavy skillet over moderate heat, stirring occasionally, until lightly browned, 4 to 6 minutes. Add mushrooms and cook, stirring, until mushrooms are tender and bacon is browned and crisp, about 4 minutes. Transfer with a slotted spoon to vegetables in saucepan.
Assemble dish: Transfer a short rib to each of 4 soup plates and keep warm in oven. Pour sauce through a medium-mesh sieve into a large bowl, pressing on and then discarding solids, then skim fat from sauce. Boil sauce, if necessary, until thickened and reduced to about 3 cups. Season with salt and pepper. Add about 2 cups sauce to vegetables (reserve remaining sauce for another use), then spoon mixture around short ribs.