Marinate the Chicken: In a bowl, whisk 3 Tbsp soy sauce, mirin, sesame oil, rice vinegar, garlic, ginger, and honey. Add chicken, toss to coat, cover, and refrigerate 30 minutes (up to 4 hours).
Cook the Rice: Combine rinsed rice, water (or broth), and salt in a pot. Bring to boil, reduce heat to low, cover, and simmer 15 minutes. Remove from heat; rest 10 minutes, then fluff.
Make the Sauce: Whisk together sauce ingredients and cornstarch slurry; set aside.
Cook the Chicken: Heat oil in a skillet over medium‑high heat. Add chicken (in batches if needed) and cook 3–4 minutes per side until golden and 165 °F inside. Pour in sauce, stir, and simmer 1–2 minutes until glossy. Remove from heat.
Prepare Vegetables: Steam broccoli and edamame 5–6 minutes. Julienne carrots and slice bell pepper; sauté 2–3 minutes or leave raw for crunch.
Assemble Bowls: Divide rice among four bowls. Top with glazed chicken, arrange vegetables, and garnish with green onions, sesame seeds, cilantro, and lime wedges.