Toss the chicken with 1 tablespoon cornstarch and 1 teaspoon soy sauce in a medium bowl; let sit 5 minutes. In a separate bowl, stir together the chicken broth, remaining 2 teaspoons each cornstarch and soy sauce and the vinegar until smooth; set aside.
Heat 2 tablespoons vegetable oil in a large nonstick skillet over high heat. Add the chicken and cook, stirring once or twice, until golden, about 3 minutes. Transfer to a plate.
Add the remaining 1 tablespoon vegetable oil to the skillet. Add the garlic and ginger and cook, stirring, until lightly browned, 30 seconds. Add the snow peas, broccolini and half of the scallions; cook, stirring, until the vegetables start softening, about 2 minutes. Stir the chicken broth mixture and pour it into the skillet. Return the chicken to the skillet and cook, stirring, until the sauce thickens and coats the chicken and vegetables, 1 to 2 minutes.
Divide the rice or quinoa among bowls and top with the stir-fry and remaining scallions. Sprinkle with toasted sesame seeds and serve with more soy sauce.