In a small bowl, mix the soy sauce, oyster sauce, cornstarch, and chicken broth until smooth. Set aside.
Heat 1 tablespoon of vegetable oil in a large skillet or wok over medium-high heat. Add the shrimp and cook for 2-3 minutes until they turn pink. Remove the shrimp from the skillet and set aside.
In the same skillet, add the remaining tablespoon of vegetable oil. Add the minced garlic and ginger, sautéing for about 30 seconds until fragrant.
Add the broccoli florets to the skillet and stir-fry for about 4-5 minutes until they are bright green and tender-crisp.
Return the shrimp to the skillet and pour the sauce mixture over the shrimp and broccoli. Stir well to combine and cook for an additional 2-3 minutes until the sauce thickens.
Drizzle with sesame oil and season with salt and pepper to taste. Serve over cooked rice or noodles.