Slice the eggplant into approximately 1 cm thick slices. Lightly salt each slice and place on paper towels. Let sit for 30 minutes. Rinse each slice with cold water and pat dry.
Brush both sides of each zucchini slice with oil. Set oven broiler to high (approximately 260°C / 500°F). Place zucchini slices on a baking rack and broil for 8 minutes. Flip and broil for another 8 minutes until nicely browned on both sides.
Wash and core the 2 yellow bell peppers. Cut into bite-sized pieces.
Lightly oil a baking dish. Layer the grilled zucchini slices in the bottom.
Brush both sides of each eggplant slice with oil. Place on the baking rack and broil at high (260°C / 500°F) for 8 minutes. Flip and broil for another 8 minutes until nicely browned on both sides.
Layer the grilled eggplant slices over the zucchini in the baking dish. Distribute the bell pepper pieces on top. Mince the garlic clove and sprinkle over the peppers. Grate Parmesan cheese over the top.
Broil at high (260°C / 500°F) for 20 to 25 minutes until the peppers are lightly browned and tender and the Parmesan is melted and golden brown.
Remove from broiler. Tear fresh basil leaves and sprinkle over the casserole. Serve immediately.