Make the Sauce: In a small bowl, combine cornstarch and water, stirring until smooth. In a medium saucepan, add the remaining sauce ingredients along with the cornstarch mixture. Stir frequently while bringing the sauce to a simmer. Cook until the sauce thickens and turns a dark brown color. Once thickened, remove from heat and let it cool completely. For faster cooling, place the sauce in the refrigerator.
Marinate the Chicken: Once the sauce has cooled, place the cubed chicken in a large ziplock bag. Pour half of the teriyaki sauce over the chicken, seal the bag, and gently shake to coat the chicken evenly. Let the chicken marinate in the refrigerator for at least 30 minutes. Reserve the remaining sauce for later.
Grill the Skewers: Preheat your grill to medium heat. Thread the marinated chicken onto the skewers. Lightly oil the grill grates, then place the skewers on the grill. Cook the chicken, turning the skewers every 3-4 minutes until the chicken is fully cooked (internal temperature should reach 165°F), about 20-30 minutes. During the last few turns, baste the skewers with the remaining teriyaki sauce.
Serve: Remove the skewers from the grill and garnish with chopped green onions and a sprinkle of sesame seeds if desired. Serve immediately.