Prepare a medium charcoal fire or light a gas grill to medium high.Meanwhile, prepare the chicken breasts. Cut through each one horizontally almost all the way through and open it like a book. Sprinkle the chicken all over with 1/2 tsp. each salt and pepper. Layer the basil and mozzarella evenly on half of each breast and then fold it closed. Wrap a slice of prosciutto around each breast to hold it closed and then brush lightly on both sides with the olive oil. Reduce the heat on the gas grill to medium. Grill the chicken breasts, covered, until they are well marked, about 4 minutes. Flip and cook, turning every few minutes, until the chicken is just firm to the touch and an instant-read thermometer inserted into the center of the breast registers 165°F, 10 to 12 minutes more. Let cool for a couple of minutes and then serve.