1Prepare the sauce: Whisk together the soy sauce, oyster sauce, honey, vinegar, Worcestershire sauce, sesame oil, garlic, ginger, and pepper in a bowl. Set aside.
2In a medium pot, cook the udon noodles according to package instructions.
3Heat the oil in a hot wok or pan over medium heat. Add the onion, garlic, and scallion whites. Cook until the aromatics are soft, about 2 minutes. Add the cabbage, carrot, and mushrooms. Cook, stirring often, until softened, about 5 minutes. Stir in the salt and pepper. Transfer the vegetable mixture to a plate.
4Add the beef to the wok. Add about 2 tablespoons of the sauce. Cook over medium heat until the beef is mostly browned, about 1 minute.
5Add the cooked noodles and vegetable mixture back to the wok. Pour over the rest of the sauce. Cook, stirring very gently so you don't break the noodles, until the sauce absorbs into the noodles, about 3 minutes.
6Garnish with the chopped green onions, and serve hot.