Heat 1 tablespoon oil in a large saucepan over medium heat. Add onion and cook, stirring occasionally, until tender and light brown, about 5 minutes. Add 1 tablespoon fresh mint (or 1 teaspoon dried), cinnamon, 1/2 teaspoon salt and pepper to taste; cook, stirring, until fragrant, about 30 seconds. Add tomatoes and water and increase heat to high. Cook, stirring occasionally, until the tomatoes begin to break down, 2 to 3 minutes. Stir in green beans. Reduce heat to medium, cover and cook, stirring occasionally, until the green beans are tender, about 12 minutes.
Meanwhile, sprinkle both sides of lamb chops with the remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Heat the remaining 1 teaspoon oil in a large ovenproof skillet over medium-high heat. Add the lamb chops and cook until browned on one side, about 2 minutes. Turn them over and transfer the pan to the oven. Roast until an instant-read thermometer inserted horizontally into a chop registers 140 °F for medium-rare, 6 to 10 minutes, depending on thickness.
Stir the remaining mint into the green bean mixture. Serve the lamb chops with the green beans.