Preheat oven to 450 degrees. Butter a 4-quart ovenproof dish. On a baking sheet, toss eggplant with 6 tablespoons oil and 1/2 teaspoon each salt and pepper. Spread in a single layer, and roast in the oven until soft and golden, 20 to 30 minutes. Transfer eggplant to prepared dish, spreading in an even layer.
In a large saucepan, warm remaining tablespoon oil over medium heat. Add onions, garlic, and ground meat; cook, stirring to prevent sticking, until meat is browned, 5 to 7 minutes. Stir in drained tomatoes, tomato paste, parsley, oregano, cinnamon, and 1/4 teaspoon each salt and pepper. Simmer, crushing tomatoes with the edge of a spoon, 15 minutes. Spread the mixture evenly over the eggplant.
Heat broiler. In a small bowl, mix ricotta, feta, egg, 1/8 teaspoon pepper, and a pinch of salt. Pour mixture over the casserole, and spread evenly to the edges. Broil until topping is browned in spots, 5 to 10 minutes. Serve hot.