Place the lamb mince, onion, garlic, eggplant, carrot, diced tomatoes, cinnamon, and oregano into a large frying pan or saucepan over medium heat.
Cook over medium-high heat for 15–20 minutes, stirring occasionally, until the lamb is cooked through, most of the liquid has reduced, and the vegetables are tender. Season with salt and pepper.
Meanwhile, boil the potatoes in salted water until tender, about 15–20 minutes. Drain well, then mash with the milk and butter until smooth.
Spoon the lamb mixture into an ovenproof baking dish and spread the mashed potato evenly over the top.
Sprinkle with the grated Parmesan cheese.
Grill for 5 minutes, or until the cheese is melted and lightly golden.
Serve warm with steamed green beans.