Place the pork butt fat-cap-side up directly on the smoker grates. If your smoker tends to run hotter on one side, position the thicker end toward the heat source. Insert a leave-in digital meat thermometer probe into the thickest part of the pork butt, making sure the tip is not touching the bone. Contact with bone gives a falsely high reading. Close the smoker and resist opening it for at least the first 3 hours. Every time you open the lid, you lose heat and smoke.
If you are doing a daytime cook: beginning at the 3-hour mark, open the smoker and spritz the pork lightly on all exposed surfaces using a spray bottle set to a fine mist. You want to dampen the surface, not wash the rub off. Spritz once every 60-90 minutes after that. If you are doing an overnight cook, skip the spritz entirely. (see Notes.)
When the internal temperature reaches 160-165°F (71-74°C), it is time to wrap. At this point the bark has set and the meat has taken on most of the smoke it will absorb. Use a wide, flat tool to slide the pork butt off the grates and onto the pan, being careful not to wipe away seasoning.
Tear off two pieces of heavy-duty aluminum foil, each about 3 feet long, and lay them out in a plus sign (+) shape. Scatter the butter pieces, brown sugar, and apple juice in the center. Place the pork butt fat-cap-side up in the center of the foil. Fold the foil up and around the pork, crimping the edges firmly to seal it as well as you can. You want the moisture trapped inside. The shiny or dull side of the foil facing in or out makes no difference.
Poke the thermometer probe through the foil back into the thickest part of the meat, away from the bone, so you can keep monitoring the temperature without unwrapping. If a tear happens during wrapping or transfer, set the pork on a fresh sheet of foil immediately, pour any escaped juice back in around the meat, and re-wrap tightly.
Return the wrapped pork to the smoker and continue cooking until the internal temperature reaches 200-203°F (93-95°C). This usually takes another 2-4 hours after wrapping. But temperature is your guide, not the clock. When the thermometer reads 200°F, insert the probe tip into 2-3 different spots in the thickest section. In a properly finished pork butt the probe will slide in and out with almost no resistance, the way a skewer goes through warm butter. If you feel any resistance at all, give it another 20-30 minutes and check again.
Before you open the smoker, stage a foil-lined sheet pan right next to it so you have somewhere to set the pork immediately if the foil tears. Using oven mitts, move the wrapped pork butt off the grates and onto your foil-lined sheet pan. Do not lift it by the foil edges. If a tear happens and juice begins leaking out, immediately wrap the foil from the sheet pan up, containing any juices within the wrap.
Carry the sheet pan to your cooler, place the wrapped pork inside, lay a clean kitchen towel or two on top, and close the lid. This is called a faux cambro and it will keep your pork butt safely above 140°F for up to 4 hours. Let it rest for a minimum of 1 to 2 hours, but 4 hours is better. Resting allows the muscle fibers to relax and reabsorb the juices that pooled during the cook.
After resting, open the foil carefully over your sheet pan. There will be hot liquid inside. Pour that liquid into a bowl or fat separator and set it aside. Wearing heat-resistant gloves or using two forks, pull the pork apart. The bone should slide out cleanly with little effort.
Discard any large clumps of fat that did not render down during the cook. Drizzle some of the reserved cooking juices back over the pulled pork and toss gently to moisten.
Serve with your favorite barbecue sauce on the side, or mix a small amount directly into the pork before serving.