Preheat oven to 400°F and line a baking sheet with parchment paper.
Arrange eggplant and potato slices on the sheet, brush with olive oil, and roast for 15 minutes until tender.
In a skillet over medium heat, brown the ground lamb with diced onion and minced garlic.
Stir in the tomato puree, cinnamon, oregano, salt, and pepper, then simmer for 5 minutes.
In a small bowl, whisk together the Greek yogurt and egg until smooth.
In a small baking dish, layer the roasted potatoes, then the lamb mixture, then the eggplant.
Spread the yogurt custard over the top and bake for 20 minutes until the topping is set and lightly browned.