Preheat oven to 325 degrees. Pat brisket dry and sprinkle generously with salt and pepper. Heat 1 Tablespoon of oil in a large oven-proof pot or Dutch oven over medium heat. Sear the meat on both sides until nicely browned, about 12 minutes total.
Transfer meat to a platter. Add 2 Tablespoons oil to the pot and then add the chopped vegetables and garlic. Sprinkle with salt and pepper and sauté over medium heat until tender and golden.
Stir in the wine, stock, tomatoes and bay leaves and deglaze the bottom of the pan by scraping the bottom with a wooden spoon. Bring to a boil and place brisket back in the pot. Cover and transfer pot to the oven. Braise the meat for 3-4 hours or until tender enough to shred with a fork.
Remove from the oven and transfer meat to a cutting board. Remove the bay leaves and discard. Remove excess fat from the meat, cut into 2-inch chunks, and shred with two forks.
If you would like a smoother sauce, pureé the vegetable mixture either directly in the pot with an immersion blender or in a food processor or blender. Otherwise you can leave it chunky. This can be done a day before and reheated in the same Dutch oven over medium heat on the stove.
Add the shredded meat back into the sauce and stir together. Serve over cooked pasta or polenta, millet-cauliflower mash, mashed potatoes or cauliflower-mashed potatoes.