Slice the chicken breast horizontally in half to get two thin cutlets. This allows it to cook quickly and stay juicy, whereas a thick breast would dry out on the outside before being cooked through. Season with salt and pepper.
Dry-toast the pine nuts in a frying pan until golden brown, then set aside.
Heat a grill pan with a tablespoon of olive oil until very hot and grill the chicken for three minutes per side until dark grill marks appear and the meat feels firm. Let it rest on a cutting board for five minutes to keep the juices in the meat. Meanwhile, grill the halved and pitted peach cut-side down for two minutes, until it starts to caramelize.
Cut the chicken into thick strips and the peach into wedges. Arrange the arugula on a plate, top with the chicken and peach, drizzle with the remaining olive oil and balsamic vinegar, and scatter the pine nuts and torn basil over the top.