Pat the lamb shanks dry and season them with salt and pepper. Heat the olive oil in a large Dutch oven over medium-high heat. Brown the shanks on all sides, then remove them to a plate.
Lower the heat to medium. Add the onion and cook until it softens. Stir in the garlic, chipotle peppers, cumin, and oregano. Cook for one minute until the spices smell fragrant.
Pour in the diced tomatoes and beef broth. Scrape the bottom of the pot to release any browned bits. Return the lamb shanks to the pot and bring the liquid to a simmer.
Cover the pot and cook on low heat for about two hours, turning the shanks once, until the meat is very tender and starting to pull from the bone.
Stir in the black beans and simmer uncovered for 15 minutes so the beans warm through and the sauce thickens slightly. Taste and adjust salt and pepper if needed.
Serve the stew in bowls with fresh cilantro and lime wedges on the side.