Make the yakisoba sauce: Get out a small bowl and add the yakisoba sauce ingredients to it. Then use a fork and mix it all together. Set this aside.
Loosen the noodles: Place the precooked noodles into a sieve and run some hot water over them for 30 seconds. Then loosen the noodles with your hands and set them aside.
Sautee the vegetables: Get out either a large skillet or a wok and pour the 2 tablespoons of oil into it. Place this over high heat, toss in the onion and carrot, and saute these vegetables for 2 to 3 minutes. Then toss in the cabbage and mushrooms and continue to saute for another minute or two. Finally, add the green onion and saute for one additional minute.
Finish the dish: Place the noodles into the skillet/wok and toss it with the vegetables. Then pour in the yakisoba sauce and saute everything for another minute or two.