In a large casserole (that has a tight fitting lid) heat 3 tablespoons oil over medium-high heat. Add the carrots, celery, and 1 onion and fry for a few minutes until softened; remove the vegetables to a plate. Season the meat with salt and pepper; add 2 tablespoons more oil to the pan, if needed, and brown the meat on both sides, about 5 minutes. Return the softened vegetables to the pan along with the bay leaf. Add enough water to just about cover the meat and bring to a boil. Lower heat, cover, and simmer gently until meat is very tender, 1 1/2 to 2 hours. Remove meat from the pot and set aside; strain the liquid and reserve.
Add another 2 tablespoons oil to the pan and put in the peppers, garlic, tomatoes, and the remaining onion; fry on medium-low heat until everything is soft and broken down, about 15 minutes. Shred the meat and add to the pan to heat through. Add some of the reserved braising liquid if the mixture becomes too dry. Serve with Black Beans and cooked white rice, garnish with cilantro leaves and queso fresco.