Place the lime in a small microwave-safe bowl and microwave until the essential oils in the skin are released. Since this lime will be hot coming from the microwave, you will need to set this aside briefly, until it cools just enough to handle. Blend the allspice, nutmeg, cinnamon, thyme, onions, scallions, pepper - together in a blender or food processor to make a pulp. Return to the lime and squeeze the juice into the blender through the feed tube. Reserve lime skins. Add the soy sauce through the feed tube and mix well.
Season chicken pieces with jerk seasoning and place them along with the lime skins in a container which you will be able to cover tightly. Pour the marinade over the chicken and rest in the refrigerator overnight (or a minimum of 4 hours). Keep tightly covered and away from other foods, as it will taint them.
Preheat oven to 350 degrees F.
Transfer the marinated chicken to a roasting pan and pour oil over the chicken. Roast covered for about 25 minutes, removing the cover and increasing the temperature to 400 degrees F for the last 5 to 10 minutes or so. When the chicken is done, the flesh will feel firm and the juices will run clear. You can also use an instant-read thermometer which should register an internal temperature of 170 degrees F.
The chicken can also be cooked on the grill. Heat grill to medium-high heat. Wrap chicken in a foil pouch in which you have shaken 1 tablespoon flour, and cook over indirect heat (on the side of the grill opposite the direct heat source) for about 15 to 20 minutes.
While the chicken is cooking, prepare the aioli. Add peaches, mayonnaise, and cilantro to a food processor and blend until smooth. Season with salt and white pepper, to taste.
Assemble sandwiches by placing sauce on roll, then chicken and top with shredded Cheddar and a sprig of cilantro.