Preheat the grill or grill pan over medium-high heat, and season the chicken breasts with salt and black pepper.
Place the chicken on the grill, cooking for about 6-8 minutes per side until the meat is golden brown and cooked through with clear juices.
While the chicken is grilling, heat a skillet over medium heat and add a tablespoon of olive oil.
Add the sliced peaches to the hot skillet, arranging them in a single layer to promote even caramelization.
Sprinkle the brown sugar evenly over the peaches and cook for about 3-4 minutes, stirring gently, until they are golden and slightly softened, releasing a caramel aroma.
Add a tablespoon of butter to the skillet and swirl to coat the peaches, allowing them to soak up the richness and develop a glossy finish.
Remove the peaches from heat and let them sit briefly so the flavors meld and juices thicken slightly.
Once the chicken reaches an internal temperature of 165°F (74°C), remove it from the grill and let it rest for a few minutes.
Serve the grilled chicken topped with the warm, caramelized peaches, spooning over any juices from the skillet for extra flavor.