Preheat the oven to 225 degrees F.
Combine all the spices for the rub and massage it into the meat. Place the rubbed pork in a roasting pan and pour the beer into the pan. Cover it with foil and roast for 6 hours. Check it periodically just to make sure everything is going along just fine, turning it over every 2 hours.
Remove the foil and roast until the pork has reached an internal temperature of 190 degrees F, about another 3 hours. Remove it from the oven, cover it loosely with foil, and let it rest for 30 minutes.
Sometime during the marathon cook time, make the sauce. In a small saucepan combine all the ingredients and bring to a boil. Taste it to make sure it is delicious. Remove the chiles. When the sauce is cool, I recommend putting it in a squeeze bottle for easy use.
Remove the bone from the pork, it should slide right out. Using your hands with dish gloves on, or with two forks, start to "pull" the pork into long shreds. Get rid of the fat and anything else that doesn't look delicious.
Serve the pork on burger buns with a drizzle of vinegar sauce to your liking and top with Broccoli Stem and Carrot Slaw.
Combine all the ingredients. Season with salt to make sure it is delicious. Let sit for at least 1 hour before serving.