Heat oven to 400° F. On 2 rimmed baking sheets, brush the eggplants with ¼ cup of the oil; season with ¼ teaspoon salt and ½ teaspoon pepper. Bake until tender, 20 to 25 minutes.
Spread 1 cup of the marinara in the bottom of an 8-inch baking dish. Top with half the eggplant, half the mozzarella, and half the Parmesan. Repeat, ending with the last cup of marinara. Top with the bread crumbs.
Bake until the cheese is melted and the sauce is bubbling, 30 to 35 minutes.
Meanwhile, whisk together the vinegar, the remaining 2 tablespoons of oil, and ¼ teaspoon each salt and pepper. Add the greens and tomatoes and toss to combine. Serve with the eggplant Parmesan.