In a small bowl, whisk together all of the marinade ingredients. Pour it into a gallon-size resealable plastic bag, add the beef, seal the bag and turn it so that all the pieces of the beef are coated in the marinade. Allow to rest at room temperature for 15 minutes.
Meanwhile, place 1 inch of water in a large sauté pan and bring to a boil over high heat. Add the broccoli, cover and steam for 3 minutes. Drain the broccoli and set aside.
Wipe the pan dry and place it over medium-high heat. Once hot, add the vegetable oil and quickly swirl it to coat the pan. Add the beef in as even a layer as possible, taking care not to overlap the pieces too much. Cook until the edges are browning, 30 seconds to 1 minute, then flip over. Add the garlic and ginger and begin to toss the beef mixture in a "stir-fry" fashion until no pink remains.
Remove the beef from the pan with a slotted spoon and place in a bowl. Whisk together the sauce ingredients and add to the pan, whisking and stirring until the sauce has thickened, about 1 to 2 minutes. Reduce the heat to low, add back the beef and broccoli to the pan and stir to coat it with the sauce. Serve immediately over white rice.
There is enough sauce to coat the beef and broccoli, but if you prefer more sauce, definitely double the sauce recipe.