Place the lamb in a heavy-bottomed pot or Dutch oven along with the onions, tomatoes, turmeric, black pepper, and cinnamon stick. Add 5-6 cups of water, ensuring the ingredients are covered by about 1½ inches.
Cover the pot and bring it to a simmer over medium-high heat. Once simmering, reduce the heat to medium and let it cook for 2 to 3 hours so the lamb is very tender.
While the base is cooking, heat the oil in a small pan over medium-high heat. Add the tomato paste and cook for about 5 minutes, stirring constantly, until it deepens in color.
Once the lamb is cooked and tender, discard the cinnamon stick. Use a slotted spoon to remove the onions and tomatoes from the pot, placing them in a bowl or measuring cup. Mash them into a smooth mixture using an immersion blender, a regular blender or a masher.
Take the lamb out of the pot, remove the bones and transfer the lamb back to the pot.
Return the mashed onions and tomatoes to the pot along with the cooked tomato paste. Add the potatoes, chickpeas, and white beans. Simmer covered over medium heat for another 30 minutes, or until the potatoes are fully cooked and tender.
Pierce the dried limes with a fork and add them to the pot. Cover and cook for an additional 10 minutes, pressing them gently with the back of a spoon to release their sour juices. Once the limes have infused their flavor, remove them from the stew and discard.
Using a slotted spoon, transfer the lamb, potatoes, chickpeas, and beans to a large bowl. Mash them into a coarse mixture using a masher.
To serve, ladle the broth into individual bowls and offer the mashed mixture on a separate platter.
Tear bread into the broth (tilit) and enjoy the stew before savoring the mashed mixture with torshi and sabzi khordan.