Soak the wooden skewers in water for 20 minutes. Preheat a gas or charcoal grill to medium-high heat (about 425°F). Clean and oil the grates.
Make the teriyaki base: In a small saucepan, whisk together the soy sauce, water, mirin, brown sugar, honey, rice vinegar, garlic, and ginger. Bring to a gentle simmer and cook 3 minutes, stirring, until the sugar dissolves.
Divide the sauce. Measure 1/2 cup into a separate small saucepan for glazing. Pour the remaining sauce into a shallow bowl and let cool 5–10 minutes.
Marinate the chicken: Add the chicken pieces to the cooled sauce in the bowl and toss to coat. Cover and refrigerate 15–30 minutes while you prepare the glaze.
Thicken the glaze: Bring the reserved 1/2 cup sauce to a simmer. Whisk in the cornstarch slurry and cook 1–2 minutes until glossy and thick enough to coat a spoon. Remove from heat and stir in the toasted sesame oil. Keep warm.
Thread and prep: Drain the chicken, discarding the marinade. Thread the pieces onto soaked skewers, leaving a little space between each piece for even cooking.
Grill: Lightly oil the hot grates with the neutral oil. Grill the skewers for 3–4 minutes per side, until lightly charred. During the last few minutes, brush with the warm glaze and turn once or twice until the chicken reaches 165°F in the center.
Finish and serve: Rest the skewers 3 minutes, then brush with a little more glaze. Shower with sliced scallions and sesame seeds and serve right away.