Adjust one oven rack 4 inches from the broiler element and a second rack to the middle position, then heat the broiler. Line a rimmed baking sheet with aluminum foil and coat with vegetable oil spray. Spread the eggplant, tomatoes, and garlic evenly over the sheet, drizzle with most of the olive oil, and season with salt and pepper. Broil until the vegetables are browned and the tomatoes have split, about 8 to 10 minutes, flipping the eggplant once halfway through.

Transfer the roasted tomatoes to a small bowl and mash them thoroughly with a fork. Mince the roasted garlic into a paste and stir it into the tomatoes along with the vinegar, salt, and pepper.

Reduce the oven temperature to 200 degrees and set a wire rack inside a clean rimmed baking sheet to keep the finished sandwiches crisp. Brush the remaining olive oil over one side of each bread slice. To build the sandwiches, place four slices oil-side down and layer with half of the mozzarella, the tomato sauce, the eggplant, the basil, and the remaining mozzarella, then top with the remaining bread slices, oil-side up. Press gently on the sandwiches to set the layers.

Heat a 12-inch nonstick grill pan over medium heat. Place two sandwiches in the pan and set a heavy Dutch oven directly on top to act as a press. Cook until the bread is golden and crisp on the first side, about 4 minutes. Flip the sandwiches, replace the Dutch oven, and cook until the second side is crisp and the cheese has completely melted.

Transfer the first batch of sandwiches to the wire rack in the warm oven. Wipe out the grill pan with paper towels and repeat the pressing and cooking process with the remaining two sandwiches. Serve warm, while the cheese is still pulling.