Make the breadcrumbs. Heat 1 tablespoon oil in a small skillet over medium heat. Add breadcrumbs and cook until golden, 5 minutes. Fold in the grated parmesan cheese and season with salt and pepper and set aside.
In a small bowl, mix ½ cup olive oil plus garlic. Brush half the garlic oil on both sides of the eggplant and season with salt and black pepper. Use the other half of the oil to brush the tomato slices on both sides.
Heat a grill or grill pan over medium high heat. Lay eggplant down and cook until browned and slightly softened, 5 minutes on both sides. Remove to a plate.
Lay tomato slices down and char slightly, 3 minutes. Transfer each tomato on top of a grilled eggplant round and return the stack to the grill. Top with a slice of cheese and cover. Cook until cheese is melted, 2 minutes.
Transfer to a serving platter and top with toasted breadcrumbs, basil, salt and pepper. Serve immediately.
Notes
Storage: Store any leftovers in an airtight container for up to 3 days. Reheat in a toaster oven, oven or grill pan to keep it firm and crisp.