With a larger cut of meat like this it really is worthwhile putting the dry rub on the night before. It is not essential, but worth the extra effort for maximum flavour.
Score any fat on the top and make some slits into the shoulder and push whole garlic cloves and rosemary sprigs into the slits. Find a rub that works well with lamb and liberally coat the joint all over. Cover over and place in the fridge over night or for as long as you can.
Place the shoulder directly onto the grills. Aiming to reach a final internal temperature of 90-93c / 195-200f. This should take around 5-7 hours however various factors such as the fat content in the lamb could vary the cooking times. A temperature probe is really an essential for larger cuts of meat.
Towards the end of the cook you could put a tray of vegetables like peppers, onions, or aubergines under the lamb while smoking to catch all the lovely juices and fat.
While the lamb is on the grill you can just let it do its thing for the first 2 hours, but after the 2-hour mark spritz or baste the lamb every 45 minutes until the required internal temperature is reached.
Mix together the vinegar, brown sugar, Worcestershire sauce, chilli powder, chopped mint, and salt into a clean spray bottle if you have one – if not a bowl and a basting brush is fine. Try to be as quick as possible when spraying and basting to avoid losing too much heat from the smoker.
Once your internal temperature has been reached and the lamb feels like it’s ready to pull apart, remove the lamb and place some kitchen foil loosely over the shoulder to rest for 15-20 minutes.
Pull the meat with a pair of forks, it should be falling away from the bone and easy to pull. Shred into soft tacos and serve with delicious sides.