Preheat your oven to 325°F (163°C).

In a large Dutch oven, heat the olive oil over medium-high heat. Season the lamb shoulder with salt and black pepper. Sear the lamb on all sides until browned, about 4-5 minutes per side. Remove the lamb from the pot and set aside.

In the same pot, add the diced onion and cook until softened, about 5 minutes. Stir in the minced garlic and cook for an additional minute until fragrant.

Pour in the pomegranate juice, balsamic vinegar, and brown sugar. Stir to combine, scraping up any browned bits from the bottom of the pot.

Return the lamb shoulder to the pot, adding the fresh rosemary sprigs and chicken broth. Bring the mixture to a simmer.

Cover the pot with a lid and transfer it to the preheated oven. Braise the lamb for 3 to 4 hours, or until the meat is tender and easily pulls away from the bone.

Once cooked, remove the pot from the oven and transfer the lamb to a cutting board. Tent it with foil and let it rest for 15-20 minutes.

To make the pan gravy, strain the liquid from the pot into a saucepan, discarding the solids. If desired, mix cornstarch with a little cold water to create a slurry and add it to the saucepan over medium heat. Bring to a simmer and cook until thickened, about 5 minutes.

Slice the lamb shoulder and serve it drizzled with the rich herb pan gravy. Garnish with fresh rosemary sprigs.