Season the chicken breasts with salt and black pepper on both sides.
Heat the vegetable oil in a skillet over medium heat.
Cook the chicken breasts for about 5 minutes on each side, or until fully cooked and no longer pink inside.
Remove the chicken from the skillet and let it rest for 5 minutes, then shred it into bite-sized pieces using two forks.
In a medium bowl, combine the Japanese mayonnaise, soy sauce, toasted sesame oil, rice vinegar, and sugar. Whisk until smooth.
Add the shredded chicken, cucumber, red bell pepper, carrot, and scallions to the bowl with the dressing. Mix well to coat all ingredients evenly.
Toast the sandwich bread slices lightly if desired.
Divide the chicken salad mixture evenly between two slices of bread.
Sprinkle toasted sesame seeds over the chicken salad on each sandwich.
Top with the remaining bread slices to form sandwiches.
Cut each sandwich in half and serve immediately.