For the Genghis Khan sauce, bring the sake and mirin (or white wine) to the boil in a small saucepan, then transfer to a food processor or blender with the remaining sauce ingredients and process to a smooth puree. Pour about the third of the sauce over the lamb and marinade in the fridge overnight. Keep the rest of the sauce in the fridge overnight before using, as it needs time for the flavours to develop.
Heat your BBQ hotplate until very hot and brush with oil. Put a little of everything onto the BBQ and cook until done to your liking, removing each vegetable as they are done and putting them onto a warmed serving plate. Pile everything onto your serving plate and top with the cooked lamb. As the lamb rests the juices will run down and flavour the vegetables. Serve with the remaining Genghis Khan sauce for dipping both the meat and the vegetables.