Preheat your oven to 400°F (200°C).
Slice the eggplant into half-inch rounds, brush with olive oil, and roast on a baking sheet for 15 minutes until tender.
While eggplant roasts, brown the ground beef in a skillet over medium heat with the diced onion and minced garlic until the meat is fully cooked.
Stir in the tomato puree, ground cinnamon, dried oregano, sea salt, and black pepper; simmer the meat sauce for 5-7 minutes.
In a small mixing bowl, whisk together the Greek yogurt and the egg until smooth to create the topping.
In a small oven-safe baking dish, layer the roasted eggplant at the bottom, followed by the meat sauce.
Pour the yogurt and egg mixture over the top, spreading it evenly to the edges.
Bake for 20-25 minutes until the custard topping is firm and lightly golden brown.