Prepare the brine by dissolving salt and sugar in 4 cups of water in a large bowl. Submerge chicken breasts completely and refrigerate for 30 minutes (not longer, as this is a quick brine).
Position oven rack in middle position and preheat oven to 425°F (218°C). Line a rimmed baking sheet with foil and lightly oil or use a non-stick spray.
In a small saucepan over medium heat, combine soy sauce, mirin, brown sugar, sake, ginger, and garlic. Bring to a simmer, stirring occasionally until sugar dissolves.
In a small bowl, mix cornstarch with cold water to create a slurry. Slowly whisk the cornstarch slurry into the simmering sauce. Continue to simmer for 2-3 minutes until the sauce thickens. It should coat the back of a spoon but still be fluid enough for brushing.
Remove sauce from heat and divide—reserve ⅓ for serving and use ⅔ for glazing during cooking.
Remove chicken from brine and pat thoroughly dry with paper towels. Discard brine. Place chicken on prepared baking sheet.
Brush chicken with a thin layer of teriyaki sauce on both sides. Place in preheated oven and bake for 10 minutes.
Lower oven temperature to 375°F (190°C). Remove chicken from oven and brush generously with teriyaki sauce. Return to oven for 5 minutes.
Remove chicken again, flip each breast, and brush with more sauce. Return to oven for another 5-7 minutes.
Apply one more layer of sauce and continue baking until internal temperature reaches 163°F (73°C) at the thickest part (about 3-5 more minutes).
Remove from oven and let rest for 5-7 minutes, loosely covered with foil. The temperature will rise to the safe 165°F during resting.
Slice chicken against the grain, drizzle with reserved teriyaki sauce, and garnish with sliced green onions and sesame seeds.