Trim any excess fat from the chicken as necessary. Pat the chicken dry with paper towels then, rub or brush the chicken with olive oil and season with salt and pepper to taste. Set aside while you heat up the grill.
Clean and oil the grill grates.
Preheat an outdoor gas grill to medium-high heat and turn off the burners on one side of the grill to create 2 zones. A direct heat zone and an indirect heat zone.
For Charcoal Grill: Clean and oil the grates. Light a chimney with enough charcoal to cover half of the grill (I fill a charcoal chimney ¾ full). When all charcoal is lit and covered with gray ash, pour out the coals on one side of the grill, leaving the other side empty. This creates 2-zones, a zone of direct heat and a zone for indirect heat. Set the clean and oiled cooking grate in place, cover the grill, and allow to preheat for about 5 minutes.
Place the chicken on the grill skin side down and cook undisturbed for about 4 to 5 minutes or until golden brown. Make sure the chicken doesn’t char.
Flip the chicken and lightly brush the top with glaze. Continue to cook covered for about 4 minutes.
Move the chicken pieces to the unlit side of grill, skin side up and generously brush with the glaze. Cover and cook for about 4 to 5 minutes.
Flip (the chicken will be skin side down) and brush with glaze. Cook covered for about 4-5 minutes. Flip again, brush with glaze and cook until the internal temperature reaches 165 degrees Fahrenheit. Remove from the grill and allow the chicken to rest for 5 to 10 minutes before serving.
Meanwhile, place the peach halves on the grill cut side down, grill for 2 minutes. Flip and brush with the reserved glaze. Grill until the peaches are soft (about 3 more minutes). Remove from the grill and serve with the chicken.