In a medium bowl, whisk together the egg white, 4 tbsp cornstarch, 1 tsp soy sauce and 1/2 tsp ginger. Stir in the chicken until evenly coated.
In a small bowl, combine the sesame oil and the remaining 1 1/2 tbsp soy sauce, 1 tbsp cornstarch and 1/2 tsp ginger. Stir in the honey and vinegar.
Heat a wok or large, heavy skillet over medium-high heat for 2 minutes. Add 1 1/2 tbsp vegetable oil and swirl to coat. Carefully add half of the chicken pieces in a single layer. Cook, turning once, until the meat is no longer pink. about 5 minutes. Using a slotted spoon, transfer the chicken to a plate. Repeat with the remaining 1 1/2 tbsp vegetable oil and chicken.
Discard the excess oil from the wok (about 1/2 tbsp). Add the scallions and garlic; stir-fry for 1 minute. Stir in the sesame oil mixture. Return chicken to the pan and cook, stirring, until the sauce is thick and the chicken is cooked through, 1-2 minutes. Serve with the broccoli and brown rice.