Whisk the soy sauce and cornstarch together in a medium mixing dish. Set aside, add the chicken broth, honey, brown sugar, rice vinegar, and sesame oil.
Heat 1 Tbsp of vegetable oil over fairly high heat in a 12-inch non-stick skillet. Dab the chicken dry with paper towels and add a single layer to the skillet.
Cook the chicken until the center registers 165, approximately 7 minutes, turning halfway through. To keep warm, switch to a sheet of aluminum foil and wrap.
Slightly reduce the temperature of the burner, heat the remaining 2 Tbsp of oil in a skillet that is now clean. Add the green onions, garlic, and ginger and sauté for 30 seconds.
Add the broccoli and water and sauté until tender, around 5 minutes, adding 1-2 tablespoons of water as necessary.
Once again, stir the soy sauce mixture and pour it into the skillet. Cook until the sauce has thickened, stirring constantly, around 2 minutes longer.
Add in the chicken and any juices that have accumulated. Toss well. Serve warm over sesame seeds sprinkled with rice.