Preheat oven to 400°F (200°C) and line a baking sheet with parchment paper.
Slice the eggplant into 1/2-inch thick rounds, brush with olive oil, and roast for 15 minutes until softened.
In a skillet over medium heat, brown the ground lamb with diced onion and minced garlic until fully cooked.
Stir in the tomato puree, dried oregano, cinnamon, salt, and pepper, then simmer for 5 minutes to meld flavors.
In a small bowl, whisk together the Greek yogurt and egg until smooth to create the custard topping.
In a small oven-safe dish, layer half the eggplant slices, followed by the lamb mixture, and the remaining eggplant.
Pour the yogurt mixture over the top, spreading evenly, and sprinkle with parmesan cheese.
Bake for 20-25 minutes until the custard topping is set and develops a beautiful golden hue.