Add chicken, scallion whites, soy sayce, sugar, 1 tsp. cornstarch, 1 tsp. salt, sherry, sesame oil, and about half the garlic and ginger to a mixing bowl. Mix the chicken around well in the mixture to coat and allow to marinate at room temperature for about 15 minutes.
Add the remaining cornstarch and 1/3 cup water.
Over high heat, add 1 tbsp. oil to a large nonstick skillet or wok. Add broccoli florets, remaining garlic, ginger, 2 tbsp. water, and 1/3 tsp. each salt and pepper. Stir fry about 2 minutes, until the broccoli is bright green. Do not overcook. Broccoli should still be crisp. Transfer this mixture to a plate.
Bring the skillet or wok back to high heat and add another 2 tbsp. oil. Add the chicken, and chili flakes if desired. Stir-fry until the chicken gets slightly brown, about 3 minutes. Add hoisin sauce, and then add the broccoli back to the pan. Stir briefly until the broccoli is reheated.
Stir in the cornstarch mixture you set aside and bring to a boil until the sauce has thickened. You can add a bit more water if needed to thin out the sauce.
Add everything to a serving bowl or platter and garnish with sesame seeds. Serve with steamed or fried rice.
You can also use the greens of the scallions to garnish if you like that raw oniony flavor.