Heat oven to 400° F. In a small bowl, combine the mint, shallot, vinegar, 3 tablespoons of the oil, and ¼ teaspoon each salt and pepper; set aside.
Heat a large skillet over medium-high heat. Season the lamb with ¾ teaspoon salt and ¼ teaspoon pepper.
In batches, brown the chops, about 2 minutes per side. Transfer to a baking sheet.
Roast for 4 to 6 minutes for medium-rare.
In a bowl, combine the potatoes and carrots. Form into 4 pancakes. Heat the remaining oil in a large nonstick skillet over medium heat.
Cook pancakes until browned, 4 to 6 minutes per side. Serve with the lamb and mint sauce.