Heat oil in a large non-stick skillet over medium-high until shimmery, about 2 minutes. Meanwhile, stir together chicken and 1 tablespoon of cornstarch in a medium bowl until chicken is coated.

Add chicken to skillet in a single layer and cook, stirring often, until crisp and cooked through, about 6 minutes. Add onions, and cook, stirring often, until translucent, about 1 minute. Add ginger and garlic; cook, stirring constantly, until fragrant, about 1 minute. Transfer chicken mixture to a plate and set aside.

Add broccoli to empty skillet, and cook over medium-high, stirring occasionally, until bright green and heated through, 2-3 minutes.

Whisk together water, soy sauce, oyster sauce, brown sugar, crushed red pepper, black pepper, and remaining 2 tablespoons cornstarch in a small bowl until smooth, about 30 seconds.

Add cooked chicken mixture to broccoli in skillet; add soy sauce mixture and cook, stirring constantly, until broccoli and chicken are well coated, and sauce thickens, about 2 minutes.