Grill any extra eggplant slices and use them for making dips or spreads (such as baba ghanoush), or add them to salads.
Preheat grill to medium-high. Brush eggplant and onion liberally with oil; season generously with salt and pepper. Brush grates with oil. Grill onion, flipping once, until softened and lightly marked in spots, about 8 minutes. Transfer to a bowl and toss with vinegar.
Grill eggplant slices until just tender on one side, about 5 minutes. Flip and top each with cheese; cook until eggplant is tender and cheese is melted, about 5 minutes more.
Meanwhile, grill rolls until lightly marked. Sandwich lettuce, eggplant, onion, and basil with rolls. Serve immediately.