Quarter the onions and remove the garlic from their skins. Add the onions, garlic, broth, and vinegar to the bottom of the slow cooker.
Set the whole lamb shoulder in the slow cooker, with the fat cap facing up. Sprinkle the top of the lamb shoulder with the Italian seasoning. If the Italian seasoning blend does NOT contain salt, sprinkle the roast with about 1/2 teaspoon of salt as well. Cover and turn the slow cooker on. Cook on Low heat for 7-8 hours, if time allows. Otherwise, set it to cook on High heat for 4.5-5.5 hours.Note: If using chops instead of a whole lamb shoulder, cook on high for 3-4 hours or low for 5.5 to 6.5 hours.
Test the tenderness of the lamb close to the end of cook time. If it's so tender that you can easily pull the meat apart with a fork, it's ready. Carefully remove the lamb from the slow cooker and set it aside on a large rimmed plate. Spoon a little liquid from the slow cooker over the meat. Cover it loosely with a piece of tin foil and let it rest.
Meanwhile, prepare the gravy. Strain all the liquid from the slow cooker into a medium sauce pan to remove the large pieces of onion and garlic. Turn to medium heat and bring to a simmer.Whisk the arrowroot powder into the simmering gravy, continuing to whisk occasionally as it thickens, for about 10 minutes. Season with salt and pepper. Note: If your arrowroot powder becomes clumpy, use an immersion blender to blend the clumps out.
Last, go back to the rested lamb. Use two forks to pull apart the pieces of the cooked lamb. Set aside the bones and any large pieces of fat. Serve the lamb with a healthy drizzle of the gravy, a little salt and pepper, and garnish with rosemary.
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Notes
Instant Pot AdaptationOptional step: Set the Instant Pot to Saute mode and add about 2 Tablespoons of olive oil. Pat the lamb shoulder dry and season with salt and pepper. Once the oil is heated and shimmering, brown the roast on all sides, about 3-4 minutes per side.Next, add the onions, garlic, vinegar, and broth to the Instant Pot. Set the lamb in the Instant Pot and sprinkle with the Italian seasonings. Secure the lid and turn to High Pressure. Set for 50 minutes, then once it's done cooking, allow for a natural pressure release (allow the steam to naturally dissipate).