In a mixing bowl combine the ground chicken, soy sauce, ginger, garlic, and sesame oil. Season with salt and pepper.
With a vegetable peeler cut the cucumber vertically into long curly strips, avoiding and discarding the seeds. Cut the carrot, again using the vegetable peeler, into strips. Combine the cucumber and the carrot in a bowl.
Heat the oil in a 9inch nonstick skillet. Shape mixture into 2 patties, each about 3/4 inch thick. Cook for about 5 to 6 minutes a side or until golden brown and cooked through. (At this point make the mayonnaise) Set each burger on a toasted bun, spread with Asian mayonnaise and top with carrot and cucumber.
Serve with Asian mayonnaise (recipe follows) and cucumber salad.