In a small bowl combine the brown sugar, salt, pepper, thyme, and paprika. Stir until well mixed.
Trim any excess fat or silverskin off the lamb. Drizzle with olive oil, then sprinkle and massage in the lamb rub until covered all over. Let sit while the smoker is set and heats up.
Set the smoker with 2 chunks of hickory at 250 degrees F (120 degrees C).
Place the lamb in the preheated smoker and smoke for 6 hours.
In a spray bottle add the water, apple juice, apple cider vinegar, and mint then shake to combine.
Spray the lamb every hour to keep moist, if it looks like it is drying out spray more frequently.
After six hours check the meat to make sure the lamb is tender. A skewer should enter with minimal resistance. If using a temperature as a guide aim for 210 degrees F (100 degrees C). This is about 10 degrees more than a pork shoulder. If not yet cooked continue until done.
Rest the lamb for 30 minutes after taking it off the smoker.
Use two forks to pull the lamb, putting aside any excess fat and the bones.
Add the BBQ sauce and mint sauce to your preferred taste, trying not to overpower the flavor of the meat. Serve immediately.