Preheat your oven to 375°F.
In a large pot, heat 1 tbsp olive oil over medium heat. Add the zucchini and eggplant, season with 1 tsp salt and pepper, and sauté for 5 minutes until softened. Remove them from the pot and set aside.
In the same pot, add 1 tbsp of olive oil and sauté the onion for 5 minutes or until translucent. Add the garlic and cook for another minute. Add the beef and cook until browned. Drain excess fat if necessary. Stir in the tomatoes, tomato paste, oregano, cinnamon, salt, and pepper. Simmer for 5 minutes or until the mixture thickens.
Stir in beef broth and rice. Bring to boil, reduce heat and simmer for 20 minutes.
When rice is finished, spread the zucchini and eggplant evenly over the top of the rice and meat mixture.
In a bowl, whisk together the Greek yogurt, eggs, Parmesan cheese, nutmeg until smooth. Pour the yogurt mixture over the layered vegetables, spreading it evenly.
Transfer the pot to the oven and bake for 30 minutes, or until the top is set and lightly golden.
When finished baking, let cool for a few minutes before slicing.