Whisk the sauce: in a medium bowl combine oyster sauce, soy sauce, chicken broth, sesame oil, light brown sugar (or honey), red pepper flakes, and cornstarch; whisk until smooth.
Optional marinate: place the cubed chicken in a zip-top bag or bowl and pour half or all of the sauce over it; refrigerate at least 30 minutes or up to overnight for more flavor.
Heat wok or large skillet over medium-high heat and add 2 tablespoons vegetable or canola oil.
Cook broccoli: when oil is hot, add broccoli florets and stir-fry 4–6 minutes until tender-crisp, then transfer broccoli to a plate.
Cook the chicken: add the chicken and remaining sauce to the hot pan in a single layer (work in batches if needed). Let the chicken sear without moving for about 1 minute, then stir and toss until no longer pink and the sauce thickens into a glaze, about 7–8 minutes.
Return broccoli to the pan and stir for about 2–3 minutes until everything is heated through and evenly coated in the sauce.
Serve immediately garnished with sesame seeds and chopped green onions.