Preheat oven to 400°F and slice eggplant into half-inch rounds.
Arrange eggplant on a baking sheet, brush with half the olive oil, and roast for 15 minutes until tender.
In a skillet over medium heat, sauté diced onion and minced garlic in the remaining oil until translucent.
Add ground lamb to the skillet, browning thoroughly while breaking it into small crumbles.
Stir in tomato puree, oregano, cinnamon, salt, and pepper, simmering for 5 minutes to meld flavors.
In a small bowl, whisk together Greek yogurt and the egg until smooth.
Layer half the roasted eggplant in a small baking dish, followed by the lamb mixture and the remaining eggplant.
Spread the yogurt mixture evenly over the top and bake for 20 minutes until the topping is set and golden.