In a heavy-bottomed pan over medium-high heat, brown the pork on all sides. Transfer the roast to the slow cooker. Return the pan to the stove.

Pour Dr Pepper into the pan and deglaze, using a spatula to scrape up all the pork bits on the bottom. Let cook for 2 minutes, scraping the bottom occasionally.

Pour the Dr Pepper over the meat in the slow cooker. Add honey, balsamic vinegar, salt, and pepper. Stir well. Cover and cook on low for 7 hours.

After the meat has cooked for 7 hours, drain off any excess liquid and shred the meat into small bits with two forks. Add the barbecue sauce to the pork and stir well. Cover the slow cooker and cook for 1 more hour.

Add more salt and pepper to taste. Fill the burger buns with pork and serve immediately.