Start by preparing all the ingredients. Peel and devein the shrimp, wash and cut the broccoli into small florets, and mince the garlic and ginger.
In a small bowl, mix together the soy sauce, oyster sauce, and sesame oil. Set aside.
Heat the vegetable oil in a wok or large skillet over medium-high heat.
Add the minced garlic and ginger to the hot oil. Stir-fry for about 30 seconds until fragrant, but do not let them burn.
Add the shrimp to the wok and stir-fry for 2-3 minutes until they are pink and opaque.
Add the broccoli florets and continue to stir-fry for another 3-4 minutes until the broccoli is bright green and tender-crisp.
Pour the sauce mixture over the shrimp and broccoli, and stir to coat everything evenly.
Add the cornstarch mixture to the wok, stirring continuously until the sauce thickens, which should take about 1 minute.
Season with salt and pepper to taste. Stir well to combine.
Remove from heat and serve immediately over cooked rice or noodles.