Purée the first 6 ingredients in a food processor or blender. Transfer to a small pot. Simmer the sauce 5 minutes, and then remove from the heat.
Preheat barbeque to medium-high. Brush the chicken with oil; season with salt and pepper. Set chicken, skin-side up, on one side of the barbecue and cook 3 to 4 minutes. Turn the chicken over and grill a few minutes on the skin-side, or until chicken achieves a nice colour. Turn the chicken breast-side up. Turn the heat off underneath it; leave the other side set to medium-high. Close the lid and cook the chicken for 25 to 30 minutes, or until cooked through. Generously baste the chicken with the peach barbeque sauce, heat through a few minutes, and then serve.