Toss the chicken with the scallion whites, half the garlic and half the ginger, the sugar, sesame oil, sherry, soy sauce, 1 teaspoon cornstarch and 1 teaspoon salt in a medium bowl. Set aside for 15 minutes to marinate. Mix the remaining 1 tablespoon cornstarch with 1/3 cup water.
Heat 1 tablespoon of vegetable oil in a large nonstick skillet over high heat. Once the oil just begins to smoke, add the broccoli, the remaining garlic and ginger and 2 tablespoons water. Season with 1/4 teaspoon salt and cook, stirring constantly, until the broccoli is bright green and crisp tender, about 2 minutes. Transfer to a plate.
Add the remaining 2 tablespoons vegetable oil to the skillet and once it just begins to smoke add the marinated chicken. Cook, stirring constantly, until the chicken begins to brown, about 3 minutes. Add the broccoli back to the skillet along with hoisin sauce and toss well. Give the reserved cornstarch mixture a good stir and add to the skillet. Bring to a boil and cook, stirring, until the sauce thickens and coats the chicken and broccoli. Remove the skillet from the heat and stir in the scallion greens. Sprinkle with toasted sesame seeds and serve over rice.
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