Whisk the stir-fry sauce by combining soy sauce, brown sugar (or honey), gochujang, rice vinegar, grated ginger, minced garlic, and toasted sesame oil in a measuring cup or small bowl until smooth.
Cook the noodles according to package directions until just al dente, then drain and set aside.
Prepare all vegetables and slice the steak before you begin stir-frying.
Heat a large wok or skillet over high heat and add 1 tablespoon sesame seed oil.
Add the carrots, red and orange bell peppers, and mushrooms (stir-fry in two batches if the pan is crowded) and cook, stirring, until they are bright and just tender, about 2–4 minutes.
Pour a few tablespoons of the sauce over the vegetables, stir to coat, cook briefly until the sauce clings to the veggies, then transfer the vegetables to a plate.
Add the remaining 1 tablespoon sesame seed oil to the hot wok. Add the sliced flank steak and let it sear without moving for about 1–2 minutes to caramelize, then toss and cook another 1–2 minutes until browned.
Add a few tablespoons of the sauce to the steak, toss to coat, then add the chopped green onions and sesame seeds and cook for about 1 minute.
Return the cooked vegetables and drained noodles to the wok, add the remaining stir-fry sauce, and toss everything together. Cook 2–3 minutes until the sauce thickens slightly and coats the noodles and beef.
Remove from heat and stir in the chopped cilantro. Divide among bowls and serve, topping with extra sesame seeds and cilantro if desired.
