Preheat grill to medium hot. In a medium mixing bowl, combine preserves, garlic, olive oil, soy sauce, dry mustard, cayenne pepper, salt, and black pepper, and mix well to combine.
Sprinkle chicken breasts with additional salt and pepper, and place, skin side down, on the grill. Cook the chicken about 10 minutes on each side before brushing the upturned side with glaze. Continue cooking chicken for another 10 to 12 minutes, turning it every 3 to 5 minutes and brushing each upturned side with glaze every time, until the chicken is cooked through. Move chicken to the oven or a cooler part of grill if it gets too dark before it is cooked through.
Place peach halves on the grill, cut side down, and grill 2 minutes. Turn, and brush the tops with glaze. Grill 3 to 4 minutes more, until the peaches are soft and the cavities fill with juices. Transfer the cooked chicken and peaches to a serving platter.
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