Preheat broiler. Combine in a small bowl the green onion, oil, vinegar, garlic, and crushed red pepper. Set aside.
Cut eggplant crosswise into 16 slices (about 1/2 inch thick). Place eggplant on the unheated rack of a broiler pan. Broil 4 to 5 inches from the heat for 8 to 10 minutes or until tender, turning and brushing once with green onion mixture halfway through broiling.
Sprinkle eggplant with mozzarella cheese, and if using, dried oregano. Broil for 1 additional minute. Sprinkle with fresh oregano, if using.