Toss chicken with its tablespoon cornstarch, tablespoon soy sauce, and tablespoon water. Stand 10 minutes while preparing ingredients, included in prep time.
Put broccoli and its water in 12-inch skillet over medium-high. Cover; steam 3 minutes until bright green and just tender. Transfer to bowl; evaporate remaining water.
Heat 1 tablespoon neutral oil. Sear half chicken 3 to 4 minutes, stirring occasionally, to 165°F; transfer to clean plate. Repeat remaining oil and chicken.
Reduce to medium; stir garlic and ginger 30 seconds. Whisk oyster sauce, sauce soy sauce, broth, and sauce cornstarch; add to skillet. Simmer 1 minute until glossy.
Return chicken and broccoli; gently toss 1 to 2 minutes until hot and evenly coated. Remove from heat; add sesame oil. Divide among 4 plates.