Season the brisket liberally with salt and pepper. Heat 1 tablespoon of the oil in a large skillet over medium-high heat. Brown the brisket on both sides, starting fatty side down. Transfer to a 5- to 6-quart slow cooker.
Add another tablespoon of oil to the skillet. Add the onions, carrots, and celery and cook until browned, about 5 minutes. Add the garlic, and cook for 30 seconds. Add the vinegar, brown sugar, ketchup, and broth and bring to a boil. Pour over the brisket, cover the cooker.
Cover and cook 4 to 6 hours on high, or 8 to10 hours on low, until the meat is very tender.
Transfer the brisket to a cutting board and skim the fat from the surface of the juices. Cut the brisket against its grain in 1/4-inch-thick slices and arrange so they are overlapping on a large platter. Surround with vegetables and spoon enough sauce over the top to moisten; serve the remaining sauce on the side.