In a small bowl, whisk water, soy sauce, rice vinegar, ginger, 2 cloves garlic, honey, brown sugar, and cornstarch-water mixture until smooth. Set aside.
Season chicken cubes well with salt and pepper. Heat a large high-walled skillet or Dutch oven over medium-high heat and swirl 1 tablespoon vegetable oil in pan.
Add chicken to pan and sear over medium-high heat until golden-brown, tossing occasionally. Add sauce mixture and cook an additional 1-2 minutes until sauce is thickened and clinging to chicken. Transfer chicken to a bowl and wipe skillet clean.
Heat 1 tablespoon vegetable oil in same pan over medium-high heat. Add broccoli and sweet pepper and saute over medium-high heat 5-7 minutes until vegetables are tender. Scrape into same bowl alongside chicken and set aside.
Heat remaining 1 tablespoon vegetable oil in same pan over medium-high heat. Add green onions and carrots and cook 3-5 minutes until crisp-tender. Add rice and remaining 2 cloves garlic and cook an additional 1-2 minutes until fragrant.
Pour chicken broth into skillet and bring to a simmer over medium-high heat. Reduce heat to low and cover pan. Allow rice to simmer on low 10-15 minutes until rice is tender, and most of liquid is evaporated.
Use a fork to fluff rice. Add teriyaki chicken and sautéed broccoli/sweet peppers to rice and fold until evenly combined. Season rice with salt and pepper to taste and top with additional green onions and sesame seeds for serving. Enjoy!