Toss chicken with its tablespoon cornstarch, tablespoon soy sauce and tablespoon water in a bowl. Let stand for 10 minutes while preparing the other ingredients; this time is included in prep.
Put broccoli and its water in a 12-inch skillet over medium-high heat. Cover and steam for 3 minutes until bright green and just tender, then transfer broccoli to a bowl and evaporate any remaining water.
Heat 1 tablespoon neutral oil in the skillet. Sear half the chicken for 3 to 4 minutes, stirring occasionally, until it reaches 165°F, then transfer to a clean plate. Repeat with remaining oil and chicken.
Lower the heat to medium and stir garlic and ginger into the skillet for 30 seconds. Whisk oyster sauce, sauce soy sauce, broth and sauce cornstarch in a bowl, pour into the skillet and simmer for 1 minute until glossy.
Return chicken and broccoli and toss gently for 1 to 2 minutes until hot and evenly coated. Remove from the heat, stir in sesame oil and divide among 4 plates.