Sprinkle salt over the chicken thighs and let them rest for 10 minutes. Pat off any excess moisture. In a small bowl, mix 1 tablespoon of water with potato starch to make a slurry.
Heat oil in a frying pan over medium heat and add the chicken. Cook until browned, then turn over and cook through. Add sugar, soy sauce, mirin, and 2 tablespoons of water, and simmer for about 1 minute. Stir in the slurry and cook stirring until the sauce thickens and coats the chicken. Cool the chicken to be able to handle.
Boil the eggs for 10-12 minutes to get perfectly hard-boiled eggs. Chop the boiled eggs with a knife and season with salt, pepper, and mayonnaise. Refrigerate until ready to use.
To assemble, place lettuce and teriyaki chicken with sauce on one slice of bread. Spread the egg salad on another slice and press them together. Wrap the sandwich tightly in plastic wrap and let it rest for 15 minutes. Cut off the crusts, then cut into thirds, remove the plastic wrap, and serve.