Pat the lamb shoulder dry. Trim any thick hard fat or loose flaps, but keep an even fat cap. Score the fat in a 1 inch crosshatch pattern, cutting through the fat but not deep into the meat.
Rub a light coat of olive oil all over the lamb. Cover the shoulder generously with the herb rub, making sure to get it into the scored fat and on all sides. Let it sit at room temperature while you fire up the smoker.
Preheat your smoker to 250 to 275°F with your choice of smoking wood. Apple, cherry, oak or pecan are all good choices. Aim for clean, thin smoke before putting the lamb on.
Place the lamb shoulder on the smoker, fat side up. Insert a temperature probe into the thickest part, avoiding the bone. Smoke for about 6 hours, spritzing lightly with the water and apple cider vinegar mixture every hour or when the surface looks dry. The bark should be dark and set and should not smear when touched. This usually happens when the internal temperature reaches about 165 to 170°F.
Transfer the lamb to a foil pan and pour about 1 cup of chicken broth into the bottom. Cover the pan tightly with heavy duty foil. Return it to the smoker and raise the temperature to 300°F. Cook for about 3 hours, or until the lamb is probe tender in several spots, around 200 to 205°F internal.
Lift the lamb out of the foil pan and wrap it tightly in foil. Place it in a warmed cooler and let it rest for at least 1 hour. You can hold it like this for up to 4 hours. Keep the pan juices separate so you can warm them later.
Once rested, pull the lamb with gloved hands. Remove the bone, large fat pockets, and any gristly bits. Warm the pan juices and mix some back through the meat to taste.
Serve the pulled lamb on rolls with BBQ sauce, load it into wraps, or use it for tacos. Fried leftovers in a hot skillet are incredible.