Season ribs with salt and pepper. In a dutch oven, heat olive oil over medium-high heat. Sear ribs, in batches if necessary, until browned all over, 10-15 minutes. Remove ribs to a large bowl and set aside.
Add carrot, celery, onion and garlic to pot; saute until browned, about 10 minutes. Add vinegar and red wine; stir with wooden spoon until all browned bits have been scraped from the pan and the bottom of the dutch oven is clean. Continue to cook until liquid is reduced by one third. Add tomatoes, chicken broth and herbs. Return ribs to pot, bring to a boil, cover pot and transfer to a preheated 300-degree oven. Cook, turning ribs once or twice, until fork tender and meat is just barely clinging to the bone, 3 to 3-1/2 hours.
Remove ribs from pot; set aside. Strain sauce, reserving cooking liquid, discarding solids. Use large spoon to remove clear fat floating on surface of cooking liquid or chill sauce until fat has hardened to remove. Cook strained and de-fatted sauce over medium-high heat until reduced and thickened, about 15 minutes.
Slice meat off ribs against the grain, into one-third-inch pieces; ribs may be served whole if desired. Serve drizzled with sauce.
Nutrition facts per serving: 1,694 calories, 151 g fat, 63 g saturated fat, 301 mg cholesterol, 13 g carbohydrates, 60 g protein, 258 mg sodium, 2 g fiber