Prep the pork. Pat the pork dry and cut it into 4 inch chunks. In a small bowl, mix the brown sugar, salt, paprika, garlic powder, onion powder, and ground mustard until combined. Sprinkle the mixture evenly over the pork pieces and rub it into the pork. Let them sit at room temperature for 1 hour.
Preheat an oven to 325 F.
Sear the pork. In a 5-6 quart dutch oven, heat the oil over medium heat. Once the oil is hot and shimmering, add the pork pieces to the dutch oven and sear them until golden brown on each side. Take care not to let the sugar in the spice rub burn.
Mix the braising liquid. While the pork is searing, whisk together the root beer, ketchup, molasses, cider vinegar, Worcestershire, dijon, brown sugar, hot sauce, and liquid smoke in a bowl or large measuring cup.
Deglaze the pan. Once the pork is browned on all sides, transfer the pork pieces to a plate and add the onions to the dutch oven. Saute them for 3-4 minutes, until they’re starting to brown. Add the chicken broth and use a wooden spoon or silicone spatula to scrape up any browned bits from the bottom of the pan. Turn off the heat and add the bourbon to the pot and stir well. Stir in 2/3 of the ketchup mixture and then add the pork back to the pot. Cover the pot and transfer to the oven.
Braise the pork. Braise the pork for 2.5-3 hours, until the pork is just shy of tender. Remove the lid and continue to cook until the pork is very tender and easily shredded with a fork. Shred the pork and pour in the rest of the ketchup mixture. Bring to a simmer over medium-low heat for 10-15 minutes, until the sauce has reduced slightly. Season to taste with salt and pepper.