Combine mayonnaise and basil in a small bowl. Using 1 tablespoon oil, lightly brush both sides of eggplant and sprinkle each slice with garlic salt. With the remaining 1 tablespoon oil, brush one side of each slice of bread.
Grill the eggplant for 6 minutes, turn with a spatula, top with cheese, and continue grilling until the cheese is melted and the eggplant is tender, about 4 minutes more. Toast the bread on the grill, 1 to 2 minutes per side.
To assemble sandwiches: Spread basil mayonnaise on four slices of bread. Top with the cheesy eggplant, red peppers, onion and the remaining slices of bread. Cut in half and serve warm.