Preheat the grill for medium heat and lightly oil the grate. Season chicken breasts with salt and black pepper.

Stir peach preserves, olive oil, soy sauce, garlic, and Dijon mustard together in a bowl until well combined. Season with salt and pepper. Reserve about 1/2 cup peach sauce for basting peaches.

Cook chicken on the preheated grill until golden brown on the bottom, 6 to 7 minutes. Flip and cook for 5 to 6 more minutes, then brush peach sauce on both sides of each breast. Cook until no longer pink in the center and the juices run clear, 4 to 5 more minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).

Meanwhile, arrange peach halves, cut-side down, on the grill. Grill for 2 minutes, then flip and brush with reserved 1/2 cup peach sauce. Continue to grill until peaches are tender, 3 to 4 minutes more.