In a bowl, whisk together soy sauce, oyster sauce, hoisin sauce, cornstarch, brown sugar, broth, and sesame oil. Set aside.
Heat oil in a large skillet over medium-high heat. Add the chicken slices and season lightly with salt and pepper. Cook for 5–7 minutes, stirring occasionally, until golden and cooked through. Remove from the skillet and set aside.
In the same pan, add a bit more oil if needed. Sauté garlic (and ginger if using) for 30 seconds, then add the broccoli. Stir-fry for 3–4 minutes until it turns bright green and is crisp-tender.
Return the chicken to the skillet and pour in the sauce. Stir continuously for 2–3 minutes until the sauce thickens and evenly coats the chicken and broccoli.
Serve hot over steamed rice or noodles for a complete meal.