In a small mixing bowl, combine soy sauce, toasted sesame oil, honey, chicken broth, and rice vinegar. Add minced garlic and grated ginger directly into the bowl.
In a separate tiny bowl, whisk together cornstarch and 1 tablespoon of water until completely smooth. Pour the cornstarch slurry into the sauce mixture and whisk until fully combined. Set aside near the stove.
Pat the chicken breasts dry with paper towels and cut into 1-inch cubes. Set aside on a clean plate.
Cut the broccoli into bite-sized florets, slice the bell pepper into thin strips, slice the carrot thinly on a diagonal, trim the snap peas, and slice the green onions (keeping white and dark green parts separate).
Place your wok or large skillet over medium-high heat and preheat for about 2 minutes. Add 1.5 tablespoons of vegetable oil and swirl to coat.
Add the chicken pieces in a single layer without overcrowding. Cook for 3 to 4 minutes without stirring until a golden crust develops. Flip and cook for another 2 to 3 minutes until barely cooked through. Transfer to a clean plate and set aside.
Return the pan to high heat and add the remaining 1.5 tablespoons of oil. Once shimmering, add the broccoli and carrots. Stir fry for 2 minutes, tossing frequently.
Add the sliced red bell pepper and white parts of the green onions. Continue stir frying for another 2 minutes until crisp-tender. Add the snap peas in the last 30 to 60 seconds.
Return the seared chicken back to the pan. Give the sauce a quick re-whisk and pour evenly over the chicken and vegetables. Toss to coat and cook over medium-high heat for 2 to 3 minutes, stirring frequently, until the sauce thickens and turns glossy.
Remove from heat and stir in the dark green tops of the sliced green onions. Spoon over steamed rice, garnish with sesame seeds and additional green onion tops, and serve immediately.