Soak the 2 leaves lettuce in cold water while you prepare the other ingredients.
2 leaves lettuce
Dust both sides of the 355 grams boneless skin-on chicken thighs with an even coating of 1 tablespoon potato starch (using a tea strainer makes this easy). Pat the chicken against each other to spread the starch around and remove any excess.
355 grams boneless skin-on chicken thighs, 1 tablespoon potato starch
Place the chicken skin-side down in a cold non-stick frying pan, put the pan over low heat, and allow the fat to slowly render out of the skin. This will take 8-10 minutes.
When the skin has crisped and the chicken is cooked halfway up the sides, flip the chicken over and turn up the heat to medium. Continue pan-frying the chicken, flipping it over every minute until it's cooked through (another 5-6 minutes). The best way to check it is to use an instant-read thermometer inserted into the thickest part of the chicken. It should read 160 degrees F (71 C).
When the chicken is cooked, remove it from the pan.
Break 2 large eggs onto opposite sides of the pan and use a spatula to fold any extra flaps of egg over towards the yolk to shape the egg so it will fit in your bun.
2 large eggs
When the white is mostly cooked through, flip the eggs over and continue frying until they reach your desired doneness. I like the whites fully set, but the yolk still runny.
Remove the eggs from the pan and use a paper towel to wipe out all of the chicken fat from the pan.
Add the 2 tablespoons evaporated cane sugar, 2 tablespoons soy sauce, and 2 tablespoons sake and let the mixture boil until it's thickened up a bit and is forming large glossy bubbles.
2 tablespoons evaporated cane sugar, 2 tablespoons soy sauce, 2 tablespoons sake
Return the chicken to the pan and repeatedly flip until the chicken is coated in a thick glaze.
To assemble the sandwich, spread 2 tablespoons Japanese mayonnaise on each of the bottom buns, dividing it between them.
2 tablespoons Japanese mayonnaise
Top this with the 1 tomato.
1 tomato
Drain and dry the lettuce and add a few layers on top of the tomatoes.
Add a piece of chicken teriyaki.
Top with a fried egg and cover with the top bun.