FOR THE SAUCE: Whisk all ingredients together in bowl; set aside.
FOR THE STIR-FRY: Whisk sesame oil, cornstarch, flour, and soy sauce together in medium bowl until smooth, then stir in chicken. In separate bowl, combine 1 teaspoon vegetable oil and ginger.
Heat 2 teaspoons vegetable oil in 12-inch nonstick skillet over high heat until just smoking. Add half of chicken mixture, breaking up any clumps, and cook, without stirring, for 1 minute. Stir chicken and continue to cook until lightly browned, about 30 seconds; transfer to clean bowl. Repeat with 2 teaspoons vegetable oil and remaining chicken mixture; transfer to bowl.
Heat remaining 2 teaspoons vegetable oil in now-empty skillet over high heat until just smoking. Add mushrooms and cook, stirring frequently, until tender and well browned, about 5 minutes. Add garlic scapes and cook until well charred, 2 to 4 minutes.
Push vegetables to sides of skillet. Add ginger-oil mixture to center and cook, mashing mixture into skillet, until fragrant, about 30 seconds. Stir ginger-oil mixture into vegetables.
Return chicken and any accumulated juices to skillet and toss to combine. Add sauce and cook, stirring constantly, until chicken and vegetables are evenly coated and sauce is thickened, about 30 seconds. Off heat, sprinkle with basil. Serve.
