Season the pork - Season all sides of the pork butt liberally with the Texas-style rub. Allow the seasoning to sweat in to the meat for about 20 minutes while the grill heats up.
Prep the grill and spritz bottle - Heat grill or smoker to 250℉ with your choice of wood or pellets. Prepare a spritzing bottle by adding apple cider vinegar to a food-safe spray bottle. See notes.
Smoke the pork butt until 170℉ - Place the seasoned pork butt directly on the grill grates, fat cap up, and smoke until the internal temperature reaches about 170℉. After the first 3 hours on the smoker, begin spritzing the sides of the pork butt with apple cider vinegar. See notes.
Prepare the braising liquid - When the pork butt is a few degrees shy of 170℉, prepare the braising liquid. Add all ingredients to a half-size deep aluminum foil pan and mix together briefly.
Continue to smoke the butt in the foil pan - When the smoked pork reaches about 170℉, transfer the pork to the foil pan with the braising liquid. Return to the smoker, and continue to smoke uncovered at 275°F until the internal temp hits 198–203°F and probes like a softened stick of butter. See notes on internal temperature.
Skim the fat from the pan juices - Once the pork butt reaches the proper temp, remove from the grill. Remove the butt from the pan and set aside. Then carefully pour the juices from the foil pan into a deep bowl or glass measuring cup. Allow the juices to cool while the pork butt rests - the fat will rise to the top of the liquid. When cooled completely, use a spoon to remove the layer of fat from the juices. Discard the fat, and set the braising juices aside.
Rest the smoked pork butt - After removing the braising liquid, return the butt to the foil pan. Wrap the foil pan with the pork in aluminum foil, and rest on the counter for 2-3 hours, or until the internal temperature of the meat drops to about 150°F. See notes on resting.
Shred and serve - Once the pork has fully rested, shred and pull the meat with your hands or two forks. Then add the reserved braising liquid, a few tablespoons at a time, and toss to coat the pulled pork evenly. You may need to reheat the liquid in the microwave first. Continue adding the reserved liquid to desired taste. Serve immediately and enjoy!