First, clip the wings and legs off the body. Cut the thighs away from the drumsticks at the joint. Remove the breast and ribs from the back, and divide the breast along the center bone. Then cut the breast halves again, cross-wise, into quarters. This makes the breast pieces roughly the same size as chicken thighs, so they cook faster and more evenly than if they were left whole.
Next, prepare your grill for indirect cooking. Friends, this is how the magic happens. Start by browning your chicken over the lit coals or gas. Once the chicken is golden, move the pieces to the cooler side — with the thicker parts facing the heat — and cook until the chicken reaches an internal temperature of 160 degrees. During the last few minutes of cooking, brush the chicken with yummy peach sauce.