In a mixing bowl, combine the soy sauce, honey, rice vinegar, sesame oil, minced garlic, and grated ginger. Whisk well to mix the ingredients.
Place the chicken breasts in a resealable plastic bag or shallow dish, and pour the marinade over them. Seal the bag or cover the dish, and marinate in the refrigerator for at least 30 minutes or up to 2 hours for best results.
Preheat the grill or grill pan over medium-high heat. Remove the chicken from the marinade and let any excess drip off.
Grill the chicken for 5-7 minutes on each side or until fully cooked and the internal temperature reaches 165°F (75°C).
While the chicken is cooking, pour the remaining marinade into a small saucepan. In a separate bowl, mix the cornstarch with the water to create a slurry. Add this mixture to the saucepan and bring to a boil over medium heat. Stir until the sauce thickens (about 2-3 minutes).
Once the chicken is cooked, brush it with the thickened teriyaki sauce. Remove from heat.
Slice the chicken and serve drizzled with additional teriyaki sauce. Garnish with sliced green onions and sesame seeds if desired.