3/4 cup soy sauce* (or mix tamari** and water in equal proportions to make 3/4 cup)
3/4 cup sake
3/4 cup mirin
4 Tbsp sugar
A 1-inch piece of ginger, grated fine
4 skinless, boneless chicken breasts, about 1 1/4 to 1 1/2 pounds total (set out for 30 minutes to come to room temp)
2-3 Tbsp sesame seeds
Mix the grated ginger, sugar, soy sauce, sake and mirin in a pot and bring to a boil. Add the chicken breasts, return to a simmer, then lower the heat to the lowest possible setting (warm if you can), on your smallest burner, and cover. The idea is to cook the chicken as gently as possible, below a simmer. Cook for 20 minutes. If you are working with somewhat large chicken breasts, you may need to cook them longer, or cut them in half before cooking.
While the chicken is poaching, toast the sesame seeds in a dry pan until they begin to brown. Move to a small bowl and set aside.
Remove the chicken breasts from the teryaki sauce, set on a plate and wrap with foil. Bring the sauce back to a boil and boil vigorously until the sauce is reduced to a syrup, about 8-10 minutes. Keep an eye on the sauce, stirring it occasionally.
To serve, slice the chicken breasts, cover with the teriyaki sauce and sprinkle sesame seeds on them. Serve with plain white rice.