Slice the chicken breasts into strips about 1/2 inch thick. Try to keep the pieces close in size so they cook at the same rate. Season lightly with black pepper. In a small bowl, whisk together the soy sauce, honey, and rice vinegar. In a separate small bowl, stir the cornstarch and cold water together until smooth. Set both aside.

Heat 1 tablespoon of the vegetable oil in a large skillet over medium-high heat until hot. Add the broccoli florets and let them cook without stirring for 2 minutes to get a little char on the edges, then stir and cook for 2 more minutes until the broccoli is bright green and just tender but still has a slight bite. Transfer to a plate.

Add the remaining tablespoon of oil to the same skillet. Add the chicken strips in a single layer and cook for 3 to 4 minutes without stirring, until the undersides are golden. Flip the strips and cook for another 2 to 3 minutes until cooked through with no pink remaining. The chicken should feel firm when pressed. If your skillet is small, cook in two batches.

Reduce the heat to medium. Push the chicken to the edges of the skillet and add the garlic and ginger to the center. Cook for 30 seconds, stirring constantly, until fragrant. Pour the soy sauce mixture into the skillet and stir everything together. Let it bubble for 1 minute.

Give the cornstarch slurry a quick stir and pour it into the skillet. Stir constantly for about 1 minute as the sauce thickens and turns glossy. It will go from thin to coating everything fairly quickly, so keep stirring so it does not clump.

Return the broccoli to the skillet and toss everything together so the sauce coats both the chicken and the broccoli evenly. Cook for 1 more minute just to heat the broccoli back through. Taste and add a little more soy sauce or honey if you want it saltier or sweeter.

Serve immediately over steamed white rice. Garnish with sesame seeds and sliced green onions if you have them on hand. The sauce will continue to thicken as it sits, so serve it right from the pan for the best texture.