Heat grill to medium. Place 4 of the tortillas on a cutting board and top with the mozzarella and tomatoes, leaving a 1-inch border.
Season with ½ teaspoon of the salt and the pepper and top with the remaining tortillas.
In a medium bowl, combine the basil, pine nuts, oil, and remaining salt; set aside.
Cook the quesadillas around the perimeter of the grill until the cheese melts and the tortillas are crisp and golden, about 2 minutes per side.
Immediately cut the quesadillas into wedges and serve with the basil salad.