Place shrimp in a large sealable container or zipped-top bag. Pour ⅓ of the marinade/sauce mixture on top, give it all a shake or stir, then allow shrimp to marinate in the refrigerator for 15 minutes or up to 8 hours. Cover and refrigerate the rest of the marinade.
Heat 1 Tablespoon of the sesame oil in a large heavy bottomed pan over medium-high heat. Place the broccoli florets in the pan. Cook for about 6 minutes, stirring occasionally, or until the broccoli is browned but silll crisp on the inside. Remove the broccoli from the pan and set it aside.
Add the remaining 1 Tablespoon of sesame oil to the pan. Place the marinated shrimp in the pan. Cook the shrimp for 1 minute, then flip it over. Quickly whisk the cornstarch into the the sauce you set aside earlier, and add it all to the pan and cook for another 1 - 2 minutes or until the shrimp is cooked through.
Stir in the broccoli and serve.