Preheat the oven to 325 degrees.
Trim excess fat from the pork and cut the meat into 4 equal pieces.
In a small bowl, combine the brown sugar, chili powder, onion powder, garlic powder, cumin, salt, and black pepper. Remove 2 tablespoons of the mixture and set it aside. Rub the remaining mixture all over the pork pieces, coating all sides. You can do this up to 24 hours in advance and refrigerate the pork until ready to cook.
Heat the oil in a Dutch oven or heavy, oven-proof pot over medium-high heat.
Add the pork to the hot oil and cook until browned on all sides and edges (the meat will brown quickly from the brown sugar, so keep a watchful eye on it).
Pour the cola all around the meat in the pan.
Cover the pan and transfer it to the oven. Bake for 3 hours.
Remove the lid and cook for 1 to 2 more hours, until the meat is tender enough to shred with a fork.
Meanwhile, to make the sauce, in a medium bowl, whisk together the ketchup, chicken broth, molasses, cider vinegar, Worcestershire sauce, liquid smoke, Dijon mustard, and reserved spice mix. Reserve for the pulled pork.
When the pork has finished cooking, shred the meat with two forks. Add the barbecue sauce and toss to coat and heat through.