Heat 1 tablespoon of vegetable oil in a large skillet or wok over medium-high heat. Add the minced garlic and ginger, and stir-fry for about 30 seconds until fragrant.
Add the broccoli florets and stir-fry for 3-4 minutes until they are bright green and tender-crisp. Remove the broccoli from the skillet and set aside.
In the same skillet, add the remaining tablespoon of vegetable oil. Add the shrimp and cook for 2-3 minutes until they turn pink and opaque.
While the shrimp is cooking, in a small bowl, whisk together the soy sauce, oyster sauce, cornstarch, and water until smooth.
Return the broccoli to the skillet with the shrimp. Pour the sauce over the shrimp and broccoli, stirring to coat everything evenly. Cook for an additional 2-3 minutes until the sauce thickens.
Drizzle with sesame oil and season with salt and pepper to taste. Serve hot over cooked rice.