Brush both sides of all the eggplant slices with ¼ cup oil, then sprinkle with salt and pepper. Let stand while you heat the grill to medium-high.
Brush the cut sides of the rolls with the remaining 2 tablespoons oil. Grill the eggplant slices, turning once, until browned and tender, 6 to 8 minutes. Transfer to a plate. Grill the cut sides of the bread until lightly browned, about 1 minute. Transfer to a large rimmed baking sheet. Reduce the grill heat to medium-low.
Spread half the sauce on the bottoms of the rolls. Top with the eggplant, then the cheese. Spread the remaining sauce on top of the cheese on each sandwich, then top with the basil leaves. Sandwich with the roll tops.
Carefully transfer to the grill and grill until the cheese melts, about 5 minutes. Serve immediately.
You can substitute Classico® Pizza Sauce for the tomato sauce. You can use regular mozzarella instead of smoked. You can use ciabatta or a baguette in place of the sandwich rolls.