Marinate the chicken: Combine chicken with 1 tablespoon cornstarch, 1 tablespoon soy sauce, and 1 teaspoon sesame oil. Let sit at least 5 minutes.
Prepare the sauce: Whisk together 1/4 cup soy sauce, rice vinegar, honey, 1 teaspoon sesame oil, 1 teaspoon cornstarch, red pepper flakes, and water. Set aside.
Prep vegetables: Cut broccoli into uniform florets. Cut pepper into similar-sized pieces. Mince garlic, grate ginger, and slice green onions, keeping white and green parts separate.
Cook the chicken: Heat a large skillet or wok over medium-high heat. Add 1 tablespoon oil and swirl to coat. Add chicken in a single layer and cook undisturbed for 2-3 minutes. Flip and cook another 2-3 minutes until golden and cooked through. Remove to a plate.
Cook the vegetables: Add remaining oil to the same pan. Add broccoli and bell pepper. Stir-fry for 4-5 minutes, stirring frequently, until broccoli is bright green and crisp-tender.
Add aromatics: Push vegetables to the sides. Add garlic, ginger, and white parts of green onions to the center. Stir for 30-60 seconds until fragrant, then mix into vegetables.
Combine and sauce: Return chicken to pan. Give sauce a quick stir, then pour over everything. Toss and cook 1-2 minutes until sauce thickens and coats everything.
Finish and serve: Remove from heat, stir in green parts of green onions, and divide among three plates. Garnish with sesame seeds if desired.