In a small bowl, whisk together soy sauce, oyster sauce, sesame oil, cornstarch, and chicken broth until smooth.
Heat 1 tablespoon oil in a large skillet or wok over medium-high heat. Add shrimp and cook for 2–3 minutes until pink and opaque. Remove and set aside.
Add the remaining oil to the pan and stir-fry broccoli for 3–4 minutes until bright green and slightly tender.
Stir in the minced garlic and ginger and cook for about 30 seconds until fragrant.
Return the shrimp to the skillet and pour the prepared sauce over the shrimp and broccoli.
Cook for 1–2 minutes while stirring until the sauce thickens and coats the ingredients.
Garnish with sesame seeds and green onions. Serve immediately with rice or noodles.