Whisk teriyaki sauce, soy sauce, sesame oil, honey, garlic, and cornstarch until smooth.
Toss chicken with flour, garlic powder, paprika, salt, and pepper until evenly coated.
Heat a large skillet with a thin film of oil over medium-high heat. Cook chicken until nearly done and lightly golden.
Pour in half the sauce and toss for 2 to 3 minutes, until the chicken is cooked through and glazed. Transfer it to a warm plate.
Add onion, bell pepper, and broccoli to the skillet. Cook until the vegetables are crisp-tender.
Add the remaining sauce and return the chicken to the skillet. Toss until everything is glossy and hot.
For the fried rice, heat butter, olive oil, and sesame oil. Add garlic and rice, then season with soy sauce.
Push rice aside, scramble the eggs in the open space, then fold in the vegetables and serve with the teriyaki chicken.