In a large skillet or wok, heat the vegetable oil over high heat. Add the sliced chicken and cook for 5-7 minutes, stirring occasionally, until browned and cooked through.
Add the broccoli florets to the skillet and stir-fry for an additional 3-5 minutes, until the broccoli is tender-crisp.
In a small bowl, whisk together the minced garlic, grated ginger, soy sauce, dark soy sauce, hoisin sauce, rice vinegar, honey, red pepper flakes, cornstarch and water.
Pour the soy sauce mixture over the chicken and broccoli and stir to combine. Cook for 1-2 minutes, until the sauce has thickened.
Add the green onions to the skillet and stir to combine.
Remove the skillet from the heat and drizzle the sesame oil over the top of the chicken and broccoli. Stir to combine.
Serve the Asian-style chicken and broccoli over cooked white rice.
