In a mixing bowl, stir the shredded chicken with the pesto until evenly coated.
Lay out the tortillas and divide the chicken mixture and mozzarella cheese between them, spreading the filling on one half of each tortilla.
Fold the tortillas in half and cook two at a time in a large skillet over medium heat for 2 to 3 minutes per side until the cheese melts and the tortillas turn golden.
While the quesadillas cook, combine the corn kernels, cherry tomatoes, and fresh basil in a second mixing bowl.
Drizzle the olive oil and balsamic vinegar over the corn mixture, season with salt and pepper, and toss gently to coat.
Slice the cooked quesadillas into wedges and serve them alongside the corn and tomato salad.