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