In large pan over medium heat, brown chicken, breaking it up into smaller pieces as it cooks. Cook all the way through, 15 minutes, stirring occasionally. Remove from pan.
Add 1 Tbsp olive oil, broccoli, and bok choy. Add remaining 2 Tbsps of oil, and stir to coat vegetables. Cook for 30 minutes, until vegetables are soft, stirring occasionally. Remove from pan.
Add sesame oil, and minced garlic. Cook for 2 minutes, until garlic is fragrant. Add remaining sauce ingredients and stir thoroughly. Cook sauce, stirring, until thickened slightly, 5 - 10 minutes.
Add chicken and vegetables and stir to coat in sauce.
Serve over white rice (optional).