In a heavy-bottomed pan over medium-high heat, brown pork on all sides. Transfer roast to slow cooker. Return pan to stove.
Pour Dr Pepper into 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 Dr Pepper over meat in slow cooker. Add honey, balsamic vinegar, salt, and pepper. Stir well. Cover and cook on low for 7 hours.
After meat has cooked for 7 hours, drain off any excess liquid and shred meat into small bits with 2 forks. Add barbecue sauce to pork and stir well. Cover slow cooker and cook for 1 more hour.
Add more salt and pepper to taste. Fill burger buns with pork and serve immediately.
