Preheat an outdoor grill for medium heat and lightly oil the grate.
Season chicken breasts with salt and black pepper; set aside. Combine peach preserves, olive oil, soy sauce, garlic, and Dijon mustard in a bowl; season with salt and black pepper. Reserve about ½ cup peach sauce for basting.
Cook chicken on the preheated grill until golden brown on bottoms, 6 to 7 minutes; flip and cook 5 to 6 minutes more. Brush peach sauce on both sides of each breast; cook until no longer pink in centers and the juices run clear, 4 to 5 minutes more. An instant-read thermometer inserted into centers should read at least 165 degrees F (74 degrees C).
Cook peach halves, cut-sides down, on the preheated grill for 2 minutes; flip and brush with reserved ½ cup peach sauce. Cook until peaches are tender, 3 to 4 minutes.