Heat grill to medium.
Finely grind the chicken breast, onion, garlic and ginger together and place in a medium bowl. Add the rice wine vinegar, soy sauce and sesame oil and mix thoroughly. Shape this mixture into 2 by 1-inch patties and let rest in the refrigerator for 5 minutes. Toast the brioche rolls, then grill the chicken patties for about 5 to 6 minutes over a medium heat. (Don't try to cook faster over high heat as the outside will be cooked and not the inside.)
To finish, place 1 patty on the bottom of each toasted bun, top with cheese, sliced tomato, onion and iceberg lettuce and cover with the bun top.