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 spray with vegetable oil spray.

Spread the eggplant, tomatoes, and garlic evenly over the baking sheet and drizzle with oil. Season with salt and pepper, then broil until the vegetables are browned and the tomatoes have split open, 8 to 10 minutes, flipping the eggplant once halfway through.

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

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 one side of each bread slice evenly with the remaining oil.

Assemble the sandwiches with the oiled sides of the bread facing out. Layer half of the mozzarella, followed by the tomato sauce, eggplant, basil, and the remaining mozzarella. Press gently on the sandwiches to set the layers.

Heat a 12-inch nonstick grill pan over medium heat for 1 minute. Place two sandwiches in the pan and set a heavy Dutch oven on top to weigh them down. 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 is completely melted, about 4 minutes.

Transfer the finished sandwiches to the wire rack in the oven to stay warm. Wipe out the grill pan with paper towels and repeat the process with the remaining two sandwiches. Serve warm while the cheese is still pulling.