Cut beef into thin strips against the grain. Chop the veggies into similar-sized pieces. Cook the Korean noodles according to the package instructions.
Preheat a wok with a little bit of oil over medium heat. Brown the mushrooms and set them aside. Brown the carrots and set them aside.
In a small bowl, whisk together all the sauce ingredients. In the wok, cook the beef slices until they form a bit of a brown crust. Pour the sauce over the beef. Let the beef simmer in the sauce for a couple of minutes.
Add the noodles, carrots, green onions, and mushrooms into the wok. Toss to combine everything together. Add spinach, stir to combine, and cook for 5 additional minutes.