Heat the oil in a large skillet over medium-high heat. Add the carrots, ½ teaspoon salt, and ¼ teaspoon pepper and cook, tossing occasionally, until tender, 5 to 7 minutes.
Add the lamb and cook, tossing occasionally, until brown, 3 to 4 minutes more.
Add the chickpeas, chard, paprika, garlic, and ½ cup water and cook, tossing frequently, until the chard is just wilted and the lamb is medium-rare, 3 to 4 minutes more. Stir in the vinegar.
Meanwhile, cook the polenta according to the package directions; stir in the butter. Serve with the lamb.