In a zipper bag, combine chicken, green onions, half the garlic, 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 and heat. Add the broccoli and red pepper; stir-fry for 1-2 minutes. Add the remaining garlic, water, and 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 again, and add 2 tablespoons oil. Add the chicken and chili flakes, if desired. Stir-fry until the chicken is no longer pink, about 3 minutes.
Add the broccoli mixture and hoisin sauce; toss to heat through.
Stir in the reserved cornstarch mixture and bring to a boil to thicken. Taste and season with salt and pepper, if you like.
Garnish with sesame seeds and serve with rice.