In a medium bowl, toss the chicken with the scallion whites, about half the garlic and ginger, the soy sauce, sugar, 1 teaspoon of the cornstarch, 1 teaspoon of the salt, the sherry, and the sesame oil. Marinate at room temperature for 15 minutes. Mix the remaining cornstarch with 1/3 cup water.
Heat a large nonstick skillet over high heat. Add 1 tablespoon of the oil and heat. Add the broccoli stems, and stir-fry for 30 seconds. Add the florets and the remaining garlic, ginger, 2 tablespoons of water, and season with 1/4 teaspoon salt, and pepper. Stir-fry until the broccoli is bright green but still crisp, about 2 minutes. Transfer to a plate.
Get the skillet good and hot again, and then heat 2 more tablespoons oil. Add the chicken and chili flakes if using. Stir-fry until the chicken loses its raw color and gets a little brown, about 3 minutes. Add the hoisin sauce, return the broccoli to the pan and 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. Mound the stir-fry on a serving platter or divide among 4 plates and garnish with sesame seeds; serve with rice.
The garlic and ginger can be chopped together in a minichopper to save time. Scallion greens can also be used to garnish, if you like that extra onion-y flavor and you want to use up the scallions. The broccoli stems are used (love all that fiber) to add textural contrast and also to help bulk up the dish, and why waste them?