In a 9- by 13-inch pan, mix onions, tomatoes, garlic, jalapeños, bay leaves, chicken broth, lime juice, vinegar, thyme, cumin, oregano, 1 teaspoon salt, and pepper. Rinse meat, place in pan, and turn to coat. Cover and chill at least 3 hours or up to 1 day, turning meat occasionally. Seal pan with foil.
Bake in a 300° oven until meat is tender enough to pull apart easily, 4 to 5 hours. Supporting with 2 slotted spoons, transfer meat to a platter. Pour pan juices through a strainer into a 12-inch frying pan or a 5- to 6-quart pan (discard bay leaves; reserve remaining vegetable mixture); skim and discard fat. Boil juices over high heat until reduced to 1 cup, 25 to 30 minutes.
When meat is cool enough to handle, in about 10 minutes, discard string. Use 2 forks or your fingers to pull meat into shreds; discard fat and connective tissue. Add meat and reserved vegetable mixture to pan with juices. Stir occasionally over medium heat until steaming, about 5 minutes. Season to taste with salt.
Nutritional analysis per 1/2 cup.