Step 1 Preheat oven to 200ºC. Process lamb in a food processor to make mince. Set aside.
Step 2 Spray a large saucepan with oil and cook onion and carrot over medium heat, stirring occasionally, for 6–7 minutes, until vegetables are soft. Add garlic and cook for 1 minute. Add mince and cook, breaking up with a wooden spoon, for 5 minutes. Drain any excess fat from pan. Add tomato paste, stirring, for 1 minute, then add tomatoes. Bring to the boil, then reduce heat to low and simmer for 10 minutes, or until thick. Season to taste and remove from heat.
Step 3 Meanwhile, spray a large chargrill pan or frying pan with oil and place over high heat. Chargrill eggplant, in batches, for 2 minutes each side, until lightly charred and tender.
Step 4 Whisk together yoghurt, egg and feta in a bowl until well combined. Season to taste.
Step 5 Spoon 1/3 of the lamb mixture into the base of a 1.5 litre (6-cup) capacity ovenproof dish. Top with 1/3 of the eggplant slices, to evenly cover the lamb. Repeat with another 2 layers of lamb and eggplant. Spread yoghurt topping over the top layer of eggplant and bake for 35–40 minutes, until topping is golden brown. Serve with steamed vegies.