Preheat the oven to 350°F (180°C). In a large Dutch oven or oven-safe dish, add the diced tomatoes, onions, carrots, and garlic, then place the thyme sprigs and bay leaves on top.
Place the whole pork shoulder on top of the vegetables. Pour the chicken broth around the meat so the liquid reaches about one-third of the height of the pork, without covering it completely. Season with salt and pepper.
Bake uncovered for 1 hour. Remove from the oven, turn the pork over, and add a little more broth if needed. Return to the oven and cook for another hour.
Remove the bay leaves, cover the pot, and continue cooking for 1 hour 30 minutes, until the pork is very tender and easy to shred.
Meanwhile, prepare the sauce. In a small saucepan, combine the red wine vinegar and cane syrup. Bring to a boil and let reduce for a few minutes until slightly syrupy. Season with salt and pepper.
Shred the pork directly in the cooking dish using two forks. Add some of the barbecue sauce and mix to coat the meat.
Fill sesame seed buns with the warm pulled pork. If desired, add cherry tomatoes, crisp lettuce, and a drizzle of vinaigrette. Serve immediately.