Preheat the oven to 375°F (190°C). Grease a 9 x 13-inch baking dish.
Make the sauce: In a small saucepan over medium heat, whisk together soy sauce, honey, rice vinegar, sesame oil, garlic, ginger, and red pepper flakes (if using). Bring to a gentle simmer.
In a small bowl, whisk cornstarch and water until smooth. Whisk it into the simmering sauce. Cook 1–2 minutes, whisking, until lightly thickened. Remove from heat.
In a large bowl (or in the baking dish), combine cooked rice, cooked chicken, broccoli, and carrots. Pour in most of the sauce (reserve a few tablespoons for finishing) and mix until evenly coated.
Spread evenly in the baking dish. Cover tightly with foil and bake 25 minutes.
Remove foil, stir gently, then bake uncovered 10–15 minutes, until hot and the veggies are tender.
Drizzle with reserved sauce. Top with green onions and sesame seeds. Rest 5 minutes before serving.