Cook the noodles al dente in a pot of boiling water following the package directions—about 2 minutes for dried chuka soba noodles or 1 minute for fresh noodles. Drain in a colander and rinse with cool water to stop the cooking. Leave in a colander in the sink to drain while you continue with the recipe.
In a bowl, whisk together the soy sauce, Worcestershire sauce, oyster sauce, hoisin sauce, chile paste, and sugar.
Heat 1 tablespoon of the oil in a wok or large skillet over medium-high heat. Add the beef and cook, stirring frequently, until browned, about 2 to 3 minutes (this will depend on how thick the strips are.) Transfer the meat to a plate or bowl.
Add another tablespoon of oil to the skillet along with the onion and carrot. Cook, stirring, until the vegetables begin to soften, about 4 minutes.
Add the mushrooms to the skillet, along with the remaining tablespoon of oil, if needed. Cook, stirring frequently, for another 2 to 3 minutes, until the mushrooms soften.
Add the cabbage and green onions along with the browned beef and the noodles. Cook, stirring to mix well, for about 2 minutes until everything is warmed through and the cabbage is beginning to wilt.
Add the sauce and cook, tossing to coat the noodles well with the sauce and mix everything together, for about 2 more minutes.
Serve hot, garnished with sesame seeds, scallions, and/or pickled ginger, if desired.