Preheat the broiler. Place the wooden skewers in a shallow container and cover them with water. Let them soak for at least 20 minutes to prevent them from burning.
Place the soy sauce, vegetable oil, ginger, garlic, mirin, brown sugar, and cornstarch in a shallow bowl and stir to combine. Add the steak cubes and toss to coat well. Let marinate in the fridge for 2 to 8 hours (if you don’t have time for this, 15 minutes at room temperature will still add nice flavor).
Lightly oil a baking sheet. Cut out and discard the stems, ribs and seeds of the bell peppers, then cut them into 1-inch squares. Slice the zucchini in half lengthwise, then cut each half into ½-inch slices. Remove the beef from the marinade, transferring the marinade to a small saucepan. Skewer the vegetables and the beef onto the wooden skewers, alternating to create a nice pattern on each (knowing there will be many more vegetables than pieces of meat. Place the skewers on the baking sheet as you go.
Broil the kebabs for about 4 minutes on one side, then turn the skewers and broil for about another 4 minutes until the teriyaki glaze caramelizes and the meat is cooked to your liking (4 minutes on each side should get you to medium rare).
Meanwhile, add any part or all of the gochujang paste to the marinade in the saucepan (if desired; depending on the level of heat you want) and bring to a boil over high heat. Reduce the heat to medium and let the marinade simmer until it is slightly thickened and reduced, about 1 minute.
You can serve this all on one big serving platter with the kebabs over the rice, or dish up individual plates with a scoop of rice and a few kebabs on top. Drizzle the spicy (or mild) simmered marinade as desired over individual portions of the rice and skewers.Serve with lime wedges for squeezing over.
Let the Kids Help:
Soak the skewers, slice the zucchini with an age-appropriate knife, and skewer the beef and vegetables (with supervision!), creating their own colorful patterns.