Toss the chicken with 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 in a bowl. Marinate at room temperature for 15 minutes. Mix the remaining cornstarch with the 1/3 cup broth or 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 1 minute. Add the florets and the remaining garlic, ginger, 2 tablespoons of water, and season with 1/4 teaspoon salt, and pepper, to taste. 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 stir-fry until the chicken loses its raw color and gets a little brown, 3 to 4 minutes. 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; serve with rice.
Serving suggestion: rice.
The garlic and ginger can be chopped together in a mini chopper to save time.
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