Heat a grill to high, an oven to 500˚F or an indoor grill pan over high heat.
Arrange eggplant slices on a kitchen towel or paper towels, salt them and let drain 10 minutes, turning once.
Combine tomatoes with onions and chopped garlic clove, and dress lightly with olive oil. Season with salt and pepper. Add olives, anchovy filets, capers and/or red pepper flakes, if using.
Brush the eggplant with olive oil on both sides then cook to tender and browned, turning occasionally and basting with oil as needed, about 10 minutes on a grill or grill pan, 12-15 minutes in the oven. Once the eggplant slices are cooked, toast rolls alongside eggplant in the oven or char on the grill. Rub charred rolls with cut garlic clove and drizzle with oil.
To assemble the sandwiches, stack eggplant with sliced mozzarella and basil leaves. Top with lots of raw tomato sauce and set bun tops in place.