Rub the chicken breasts with the jerk seasoning, salt, and pepper. Pour the olive oil into a medium sauté pan, and place it over high heat. When the oil is almost smoking, add the chicken, turn the heat down to medium high and continue searing on both sides until the chicken is golden brown and thoroughly cooked, about 4 to 5 minutes. Transfer the chicken from the pan to a cutting board and let the chicken cool. When it’s cool enough to handle comfortably, slice the chicken breasts into 1/4-inch strips.
Cover a bamboo mat or sheet of foil with plastic wrap. Place a sheet of nori on top. With water-moistened hands, spread 1 cup of the cooked rice evenly on top of the seaweed leaving about a quarter inch border at the top and bottom edges. Press firmly. Leaving the plastic wrap in place on the mat, turn the sushi layer over so that the seaweed faces up and the rice faces down.
Arrange one quarter of the chicken strips, mango and avocado slices, and cucumber on top the seaweed so the ingredients form a neat narrow row that aligns with the edge of the nori square closest to you. Roll the bamboo mat forward (making sure the plastic wrap stays with the mat), pressing the ingredients inside the cylinder-shaped sushi. Press the bamboo mat firmly with both hands to shape the sushi, then remove the mat. Repeat the process with remaining ingredients to make three more rolls. Serve with soy and wasabi or with Avocado Tzatziki.
Cut the sushi rolls into 1 inch rounds with a sharp serrated knife moistened with water. Serve with Pickapeppa sauce and pickled ginger.