Heat oven to 350 degrees. Bring a large pot of salted water to boil. Cut a deep cone-shaped incision into bottom of each cabbage; remove cores. Add the whole cabbage and boil until leaves soften, about 7 minutes, turning them around in the water occasionally. Transfer to a colander and drain, cut side down. Gently pull off 8 of the large, outer leaves from each head. Trim thick ribs from leaves and set aside. Cut remaining cabbage heads in half and thinly slice the remaining leaves. In a large saute pan over medium high heat, cook the ground beef, breaking it into small pieces as it cooks, until 75% cooked and still pink. Drain and discard oil. Transfer the meat to a large bowl and reserve. Wipe the pan clean and the heat oil to medium. Gently saute the onion, carrot, celery, and garlic. About 8 minutes. Raise the heat to medium-high and add the cabbage and eggplant. Season with salt and pepper, cook, stirring often for about 5 minutes, or until the cabbage is well wilted. Add the broth and bring to a boil. Transfer the contents of the pan to the bowl with the reserved meat and stir well. Stir in the bread crumbs and parsley. Butter a rimmed baking sheet. Place 1 cabbage leaf on a work surface. Spoon about ½ cup of the stuffing onto center of leaf. Fold bottom of leaf over filling. Fold in sides. Continue rolling to enclose filling. Repeat with remaining cabbage leaves. Arrange packages seam side down on prepared baking sheet. Brush tops with olive oil. Cover baking sheet with foil. (Can be prepared 1 day ahead. Refrigerate.) Bake packages until heated through, about 25 minutes.