In a small bowl, whisk together chicken broth, brown sugar, soy sauce, oyster sauce, 1 teaspoon sesame oil, and cornstarch until smooth; set the sauce aside.
Heat 1 tablespoon sesame oil in a large wok or non-stick skillet over medium-high heat.
Season the chicken pieces with kosher salt and black pepper, then add to the hot pan and cook, stirring occasionally, about 6 minutes until golden and no longer pink; remove the chicken and set aside.
Add the remaining 1 tablespoon sesame oil to the wok, then add minced garlic and ginger paste and cook, stirring, about 30 seconds until fragrant.
Add the broccoli florets and stir-fry about 4 minutes until tender-crisp and lightly browned; remove the broccoli and set aside.
Return the wok to medium-high heat, pour in the prepared sauce, and cook 1–2 minutes until it begins to thicken.
Return the cooked chicken and broccoli to the wok, toss to coat evenly with the sauce, and heat through about 2–3 minutes.
Serve immediately over rice and optionally garnish with sesame seeds and red pepper flakes.