Combine sliced steak with 1 tablespoon of dark soy sauce and 2 tablespoons of mirin. Set aside to marinate while working on the sauce.
In a small bowl whisk together the ingredients for the stir-fry sauce: the remaining 3 tablespoons soy sauce, oyster sauce, ginger paste, garlic paste, and brown sugar.
Spoon a couple of tablespoons of the beef broth into the cornstarch and stir to dissolve the cornstarch.
Mix both the cornstarch mixture and the remaining beef broth into the rest of the stir-fry sauce.
Drain the marinated steak and discard juices.
Heat the oil in a skillet or wok over medium-high heat until the oil is nearly smoking and a drop of water in the pan sizzles immediately.
Drop the drained meat into the pan and stir-fry for 2 to 3 minutes until the meat is browned on both sides.
Remove the meat from the pan and set it aside.
Add a thin layer of water to the bottom of the pan, just a few tablespoons to steam/simmer the broccoli in. You can use a clean pan or the pan you’ve just used to cook the meat.
Add the broccoli florets to the skillet and bring to a simmer. Simmer over medium-high heat, tossing the broccoli a few times until it begins to soften and turns vibrant green, about 3 to 4 minutes.
Add about half of the prepared sauce to the broccoli and bring to a simmer. Stir and simmer 1 to 2 minutes longer until the sauce thickens.
Add the cooked meat back into the pan and add as much of the remaining sauce as you’d like. This recipe makes a very saucy dish.
Heat the meat through and simmer long enough to thicken any additional sauce you have added.
Garnish with sliced green onions and toasted sesame seeds, if desired.
Serve over white rice or fried rice