Preheat the oven to 160° fan. Put the lamb in a large roasting tin or casserole dish and rub it with olive oil, salt and pepper.
Using a thin sharp knife, make a few cuts in the lamb, as deep as you can, and fill the holes with pieces of rosemary and garlic.
Place the onion, the halved head of garlic and the remaining rosemary in the bottom of the roasting tin and add the water.
Cover with a lid or foil and roast for 3 hours until soft and falling apart.
When there is an hour left for the lamb, prepare the potatoes. Cut each potato thinly almost all the way through. Mix the butter, garlic, rosemary, salt and pepper in a small bowl. Brush over the potatoes and season well. Roast for 50–60 minutes until golden and crispy, basting halfway through.
Raise the temperature to 200° fan and remove the lid from the lamb. Roast for a further 20–30 minutes until the meat is golden and the potatoes are crispy.
Remove the lamb from the roasting tin and transfer to a plate. Cover loosely with foil and let rest for 20 minutes.
While the lamb is resting, make the mint sauce. Mix the mint, sugar and boiling water in a small bowl. Mix well and add the vinegar and season.
Serve the lamb and potatoes with a mint and broccoli sauce.