Step 1

PreparationBraise short ribs: Put oven rack in lower third of oven and preheat oven to 250°F.

Step 2

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.

Step 3

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.

Step 4

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.

Step 5

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.

Step 6

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.

Step 7

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.

Step 8

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.

Step 9

Cooks' Note: