Remove the lamb from the refrigerator and allow it to reach room temperature.

Preheat the oven to 460°F. Thoroughly dry the lamb, then rub it with olive oil, salt, and pepper, and let it sit for 15 minutes.

Place the shoulder skin-side up in a baking dish or roasting pan and roast for 15–20 minutes, until it develops a golden-brown color. 

Reduce the heat to 300°F, pull out the roasting pan, add 1 cup of chicken broth to the pan, and cover tightly with a layer of parchment paper and two layers of foil. Return to the oven and cook for 3–4 hours. Every hour or so, remove the cover and baste it with the sauce at the bottom of the pan, then cover tightly again and continue cooking.

Meanwhile, prepare the apricot sauce: Heat 2 tablespoons of olive oil in a saucepan over medium-high heat. Add the shallots, season with salt and pepper, and sauté until translucent, 5-7 minutes.

Add the dried apricots and sauté for another 3–4 minutes. Stir in the thyme and wine, and allow the alcohol to evaporate, about 5-7 minutes. Add 2 cups of broth, bring to a boil, then reduce the heat to medium-low and simmer gently, uncovered, for 25–30 minutes, until the apricots are completely soft.

Using an immersion blender, blend the sauce to the desired consistency, either slightly chunky or smooth, then stir in the lemon zest and juice. Taste and adjust seasoning with additional salt, lemon juice, or honey as needed. 

You’ll know the lamb is ready when it can be easily pulled from the bone with a fork. Test this from the bottom of the shoulder so as not to affect its appearance. When the lamb is ready, remove the cover and roast uncovered for about 10 minutes, until the skin develops a deep golden color. Remove the lamb from the oven and let it rest, covered with its parchment paper, for 15 minutes. 

To serve, carefully transfer the lamb from the roasting pan to a platter. At the table, use tongs or two forks to pull the meat and serve onto individual plates. Serve with warm apricot sauce on the side.