Heat a wok or large pan over medium heat and add the oil. Add sliced onions and minced garlic, and stir-fry until fragrant and softened, about 1-2 minutes.
Add cubed tofu to the wok, with the soy sauce, and stir-fry until the tofu begins to brown slightly, ensuring the soy sauce is absorbed.
Add the sliced mushrooms, shredded cabbage and julienned carrots to the wok, cooking for another 2-3 minutes until the vegetables soften but still retain some crunch.
Add the cooked udon noodles to the wok along with the prepared sauce. Toss everything gently, ensuring the noodles are evenly coated with the sauce while being careful not to break them.