Generously brush both sides of the eggplant slices with olive oil. Place on a baking sheet and broil until lightly browned on both sides. Transfer to a large plate and set aside. Heat the vegetable oil in a large skillet. Add the onion, carrots, and garlic and stir over medium heat until the onion is tender. Stir in the ground meat and cook until no longer pink. Add the tomatoes, basil, and oregano and bring to a gentle bubble. Reduce the heat and cover the pan. Simmer for 10 minutes. Season with salt and pepper and increase the heat. Cook, uncovered, at a gentle bubble for 30 minutes or until thickened. Preheat the oven to 350 degrees. Generously butter a shallow, rectangular baking dish. Spoon in enough tomato sauce to cover the bottom of the pan. Layer on half of the eggplant slices. Spoon the ricotta cheese over the eggplant and scatter on half the feta cheese and half the Parmesan cheese. Ladle half of the remaining tomato sauce over the cheese. Layer on the remaining eggplant slices and then sprinkle with the remaining feta cheese and parmesan cheese. Ladle the remaining tomato sauce over the cheese and cover the dish with aluminum foil. Bake for 30 minutes. Uncover the dish and bake for 15 to 20 minutes or until the sauce is bubbling.