Heat the broiler. Place the eggplant slices on a rack in a broiler pan. Brush the eggplant slices with 1 tablespoon oil.
Broil the eggplant with the top of the eggplant 7 inches from the heat for 5 minutes or until the eggplant is lightly browned on both sides, turning the eggplant over once halfway through broiling and brushing with 1 tablespoon oil.
Place the eggplant and arugula into a large bowl. Add 1 tablespoon oil and the vinegar and toss to coat. Divide the eggplant, arugula and cheese among the rolls.
Heat the panini maker. Brush the tops of the sandwiches with the remaining oil. Add the sandwiches in batches, oil-side down, and cook for 5 minutes or until lightly browned and the cheese is melted.
Alternate Preparation: If you don't have a panini maker, you can use a grill pan or a skillet. Heat the pan over medium heat. Add the sandwiches in batches and cook for 5 minutes, pressing down on the sandwiches with a spatula, until they're lightly browned on both sides and the cheese is melted, turning over once halfway through the cooking time. You can also weight the sandwiches down with a foil-wrapped brick or a cast-iron skillet instead of pressing down with a spatula.