Rub the chicken breasts with the jerk seasoning, salt, and the black pepper. In a medium saute pan, add the olive oil, and place it over high heat. When the oil is almost smoking, add the chicken breasts, 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 per side. 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 1/4-inch border at the top and bottom edges. Press the rice firmly on the nori. 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 1/4 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, and then remove the mat. Repeat the process with remaining ingredients to make three more rolls. Place the sesame seeds on a plate. Roll the sushi in the sesame seeds until coated. Using a sharp serrated knife, moistened with water, cut the sushi rolls into 1-inch rounds. Serve the rolls with soy sauce and wasabi or with the Avocado Tzatziki.
Serve with Pickapeppa sauce and pickled ginger. .