Preheat your oven to 400°F and line a baking sheet with parchment paper.
Slice the eggplant into 1/2-inch thick rounds, brush both sides with olive oil, and sprinkle with a pinch of salt.
Roast the eggplant for 15-20 minutes until softened and slightly browned, then remove from the oven and lower the heat to 375°F.
While eggplant roasts, sauté the diced onion and minced garlic in a non-stick skillet over medium heat until translucent.
Add the ground lamb to the skillet, breaking it up with a spoon, and cook until browned.
Stir in the tomato puree, oregano, cinnamon, and pepper; simmer for 5 minutes until the sauce has thickened.
In a small bowl, whisk together the Greek yogurt, egg, and nutmeg until completely smooth.
In a small oven-safe baking dish, layer half of the eggplant slices, followed by the lamb mixture, and then the remaining eggplant.
Pour the yogurt custard over the top, spreading it evenly to the edges.
Bake for 20-25 minutes until the topping is set and beautifully golden brown.