Heat 2 tablespoons olive oil in a large heavy-bottomed Dutch-oven or roasting pan over medium-high heat. Season brisket with 1 1/2 teaspoons salt and 1/4 teaspoon pepper and add to pan. Cook brisket, turning, until browned on all sides, 4 to 5 minutes per side. Transfer to a plate; set aside.

Reduce heat to medium-low and add remaining tablespoon olive oil, along with onions and garlic. Cook, stirring frequently, until browned and soft, about 25 minutes. Add wine and increase heat; bring to a boil. Let boil for 2 minutes, scraping up browned bits from bottom of pan using a wooden spoon. Add vinegar, 4 carrots, 2 ribs celery, and tomatoes; stir to combine.

Return brisket to pan; spoon over some of the cooking liquid and vegetables. Cover with parchment paper-lined foil and transfer to oven. Cook for 1 1/2 hours; carefully turn meat and continue cooking until very tender, about 1 1/2 hours more.

One hour before brisket is finished cooking, bring a large pot of salted water to a boil. Add remaining 4 carrots and cook until tender, about 10 minutes. Remove from boiling water using a slotted spoon and set aside. Add remaining 4 ribs celery and cook until tender, about 10 minutes. Repeat process first with turnips, followed by rutabagas; set vegetables aside.