In a small bowl, whisk together soy sauce, chicken broth, oyster sauce, and brown sugar to create the sauce base.
In a separate small cup, mix cornstarch and water to create a slurry.
Heat 1 tablespoon of vegetable oil in a wok or large skillet over high heat.
Add broccoli florets and 2 tablespoons of water, cover and steam for 2 minutes until bright green and tender-crisp; remove from pan.
Add remaining 1 tablespoon of oil to the pan and stir-fry chicken strips until browned and cooked through.
Add minced garlic and ginger to the chicken and sauté for 30 seconds until fragrant.
Pour the sauce base into the pan and bring to a simmer.
Stir in the cornstarch slurry and cook until the sauce thickens and becomes glossy.
Return the broccoli to the pan and toss to coat everything evenly in the sauce.
Drizzle with sesame oil and serve immediately.