Start by preparing all the ingredients. Mince the garlic and ginger, and cut the broccoli into florets.
In a small bowl, combine the soy sauce, oyster sauce, cornstarch, chicken broth, and sesame oil. Mix well to create the sauce.
Heat the vegetable oil in a wok or large skillet over medium-high heat.
Add the minced garlic and ginger to the wok, stirring for about 30 seconds until fragrant.
Add the shrimp to the wok and cook for about 3-4 minutes, or until they turn pink and opaque. Remove the shrimp from the wok and set aside.
In the same wok, add the broccoli florets and stir-fry for about 2-3 minutes until they are bright green and tender-crisp.
Return the cooked shrimp to the wok. Pour the sauce mixture over the shrimp and broccoli, stirring well to coat.
Cook for another 2-3 minutes or until the sauce thickens and everything is heated through. Season with salt and pepper to taste.
Serve hot over cooked rice or noodles if desired.