Sprinkle the brisket on both sides with salt and pepper. Heat the oil in a large skillet over medium-high heat. Add the brisket (slice it in half lengthwise if too large for the skillet) and cook until browned on both sides, 4 to 6 minutes per side. Remove from the pan and set aside.
Stir the tomato paste into the pan and cook until heated through, about 2 minutes. Add the mustard and 1 cup of the vegetable broth; scrape the bottom of the pan with a wooden spoon to remove any browned bits. Carefully transfer the liquid to a slow cooker, along with the onions, carrots and garlic. Then add the brisket, vinegar, Worcestershire, bay leaves, and the remaining 1 cup broth. Cook on low until the brisket is very tender, about 8 hours.
Remove the brisket to a platter and let rest for 20 minutes. Meanwhile, strain the cooking liquid, season it with salt and pepper, and keep warm.
Slice the brisket against the grain and serve it with the sauce.
Recipe courtesy of Trisha Yearwood