Make the teriyaki sauce: combine soy sauce, brown sugar, honey, rice vinegar, and sesame oil in a small saucepan. Add garlic and ginger. In a small bowl, whisk cornstarch with cold water until dissolved.
Bring the sauce mixture to a gentle boil over medium heat. Stir in the cornstarch slurry, reduce heat to low, and simmer 3-4 minutes until thickened. Reserve one-third for glazing and serving.
Preheat oven to 400°F (200°C). Line a large rimmed baking sheet with foil or parchment paper and lightly coat with oil or cooking spray.
Prep vegetables: cut broccoli into florets, slice bell pepper into 1-inch pieces, slice carrots into thin rounds, and cut red onion into wedges.
Toss vegetables with olive oil, salt, and pepper in a large bowl. Add half of the teriyaki sauce and toss to coat evenly.
Pat chicken thighs dry with paper towels and season lightly with salt and pepper. Arrange vegetables in a single layer on the pan, place chicken in the center, and brush with remaining coating sauce.
Bake for 20-25 minutes until chicken reaches 155°F to 160°F (68°C to 71°C). Vegetables should be bright and slightly caramelized at the edges.
Switch oven to broil on high. Brush chicken with reserved sauce and broil 2-3 minutes until glossy and caramelized, watching closely. Internal temperature should reach 165°F (74°C).
Rest chicken 3-4 minutes. Drizzle with remaining reserved sauce, sprinkle with sesame seeds and green onions, and serve over steamed rice.