Preheat oven to 400°F and line a baking sheet with parchment paper.
Slice the eggplant into 0.5-inch rounds, brush with olive oil, and roast for 20 minutes until tender.
While roasting, sauté the ground beef, onion, and garlic in a skillet over medium heat until the beef is browned.
Stir in the tomato puree, cinnamon, oregano, sea salt, and black pepper, then simmer for 10 minutes.
In a small bowl, whisk together the Greek yogurt, egg, and parmesan cheese until the mixture is smooth and creamy.
In a small oven-safe dish, layer half the eggplant, all the beef mixture, the remaining eggplant, and spread the yogurt custard on top.
Bake at 375°F for 25 minutes until the topping is golden brown and bubbly.