Preheat your oven to 325°F (163°C).

In a large, heavy-bottomed Dutch oven or oven-safe pot, heat the olive oil over medium-high heat. Season the lamb shoulder with salt and pepper. Sear the lamb on all sides until it is 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 and translucent, about 5 minutes. Add the minced garlic and cook for an additional 1-2 minutes until fragrant.

Pour in the pomegranate juice, balsamic vinegar, honey, and thyme. Stir to combine and bring to a simmer.

Return the lamb shoulder to the pot, nestling it into the sauce. Add the broth, ensuring the lamb is mostly submerged.

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 apart with a fork.

Once the lamb is done, remove it from the oven and transfer the lamb to a cutting board. Let it rest for 15-20 minutes before slicing.

Meanwhile, place the pot with the braising liquid over medium heat on the stove. Bring the sauce to a simmer and cook for 10-15 minutes, allowing it to reduce slightly and thicken. Adjust seasoning with salt and pepper as needed.

To serve, slice the lamb shoulder and arrange it on a platter. Drizzle the velvety pan gravy over the top and garnish with fresh thyme sprigs.