In a mixing bowl, combine the cubed chicken with 1/2 tablespoon cornstarch and a pinch of salt; if needed, add a splash of water so each piece is evenly coated. Let sit while you prepare the other ingredients.
Bring a pot of water to a boil. Blanch the broccoli florets for 1 minute, drain in a colander, and set aside.
In a wok or large skillet over high heat, heat 2 tablespoons oil until hot. Add the minced garlic and stir-fry briefly until fragrant and lightly golden.
Add the marinated chicken to the wok and stir-fry on high heat until the pieces are mostly cooked through, about 2–3 minutes.
Stir together 1 tablespoon soy sauce, 1 tablespoon oyster sauce, 1/2 teaspoon sesame oil, 1 teaspoon sugar, 1 teaspoon cornstarch, and 1/2 cup water; add 3 dashes white pepper. Pour the sauce into the pan, reduce heat to low, and simmer briefly until the sauce thickens and the chicken is cooked through.
Add the blanched broccoli to the pan, toss until everything is evenly coated and heated through, then transfer to a serving dish and serve immediately.