In a medium bowl, toss the chicken, scallion whites, half the garlic, half the ginger, soy sauce, sugar, 1 teaspoon cornstarch, salt, sherry and sesame oil. Marinate at room temperature for 15 minutes.
Mix the remaining cornstarch with 1/3 cup cold water.
Heat a large nonstick skillet over high heat. Add 1 tablespoon oil. Add the broccoli and carrots. Stir-fry for 2 minutes. Add the remaining garlic and ginger and 2 tablespoons water, season with salt and pepper. Stir-fry until the broccoli is bright green but still crisp, about 2 minutes. Transfer to a plate.
Heat the skillet to high again. Add remaining 2 tablespoons oil. Add the chicken mixture. Stir-fry until the chicken loses its raw color and gets a little brown, about 3 minutes.
Add the hoisin sauce and broccoli mixture to the pan. Toss to heat through.
Stir in the reserved cornstarch mixture and bring to a boil to thicken. Add more water if need to thin the sauce, if necessary. Taste and season with salt and pepper, if you like.
Serve with rice, if desired.
