Rub the lamb shoulder evenly with olive oil, salt, and black pepper.
Using a thin, sharp knife, make 12 deep incisions all over the lamb, ensuring you do not pierce completely through the bottom.
Stuff small sprigs of rosemary and the slivered garlic into the incisions (using a chopstick can make this easier).
Scatter the quartered onion, halved garlic head, and remaining rosemary across the bottom of a roasting pan. Place the prepared lamb on top. Pour the water into the bottom of the pan around the lamb. Cover securely with a lid or tightly with a double layer of aluminum foil, then transfer to the oven.
Slow roast (covered): Roast covered for 5 hours.
Brown (uncovered): Remove the foil and check that there is still liquid remaining in the pan. If dry, add 3/4 cup of water to prevent the onion and garlic from burning. Increase the oven temperature to 220°C (425°F) and roast uncovered for an additional 20 to 30 minutes, until the skin is well-browned and crisp.
Check doneness: Test the meat with two forks; it should pull apart effortlessly. If it does not, cover and return to the oven at 180°C (350°F / 160°C fan) until fork-tender.
Rest: Transfer the lamb to a serving platter. Tent loosely with foil and cover with a couple of clean kitchen towels. Let rest for at least 20 minutes and up to 2 hours before carving or pulling apart.
Gravy Instructions
Tilt the roasting pan and spoon out excess fat, leaving approximately 2 tablespoons of fat along with the pan juices.
Set the roasting pan directly over the stovetop on medium-high heat. Add the flour and stir continuously into the fat for about 30 seconds to cook out the raw flour taste.
Gradually pour in the beef broth, whisking constantly to combine. Use a potato masher to crush the softened onion and garlic, pressing the garlic out of its skins into the sauce.
Simmer for 1 to 2 minutes until slightly reduced and approaching your desired consistency (it will continue to thicken slightly as it cools). Remove from heat.
Season to taste with salt and pepper. Strain the gravy through a fine-mesh sieve into a bowl, pressing firmly on the solids to extract all flavor, then pour into a gravy boat to serve.