Begin by prepping your ingredients. Peel and devein the shrimp if not already done, and chop the broccoli into florets. Mince the garlic and ginger.
In a small bowl, mix the soy sauce, oyster sauce, cornstarch, chicken broth, and sesame oil. Set this sauce mixture aside.
Heat the vegetable oil in a wok or large skillet over medium-high heat. Once hot, add the minced garlic and ginger, cooking for 1 minute until fragrant.
Add the shrimp to the wok and stir-fry for about 3-4 minutes, or until they turn pink and opaque.
Toss in the broccoli florets and stir-fry for an additional 3 minutes until bright green and tender-crisp.
Pour the sauce mixture over the shrimp and broccoli, stirring to coat everything evenly. Cook for another 2 minutes, allowing the sauce to thicken slightly.
Season with salt and pepper to taste, and remove from heat.
Serve the shrimp and broccoli stir-fry over cooked rice or noodles, if desired.