Step 1
In a medium bowl, toss the chicken with the sliced white onions, sugar, sesame oil, sherry, soy sauce, about half the garlic, half the ginger, 1 teaspoon of the cornstarch and 1 teaspoon salt. Marinate at room temperature for 15 minutes. Mix the remaining 1 tablespoon cornstarch with 1/3 cup water in a small bowl and reserve

Step 2
Heat a large nonstick skillet over high heat. Add 1 tablespoon of vegetable oil and heat. Add the broccoli stems and carrots and stir-fry for 30 seconds. Add the florets and the remaining garlic and ginger, 2 tablespoons water, 1/4 teaspoon salt and some black pepper. Stir-fry until the broccoli is bright green but still crisp, about 2 minutes. Transfer to a plate.

Step 3
Make the skillet good and hot again, and then heat the remaining 2 tablespoons vegetable oil. Add the chicken and red pepper flakes if used. 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.

Step 4
Mound the stir-fry on a serving platter or divide it among 4 plates and garnish with sesame seeds; serve with rice.