For the sauce: In a small bowl, mix all sauce ingredients together until combined. Set aside.
Prepare the Noodles: If using frozen udon noodles, soak them in hot water for 1-2 min, carefully separating the noodles using chopsticks, then drain and rinse well under cold water. Set aside until ready to use. If you're using shelf-stable packaged noodles, you don't have to soak them but it's helpful to gently separate and loosen the noodles.
Stir Fry the Veggies: In a large pan or wok, heat 1 tbsp oil over medium-high heat. When the pan is hot, add the carrots, cabbage and mushrooms. Cook for 5-6 min, stirring often, until tender.
Add Noodles: When the veggies are tender, add the green onions (reserve some green onions for topping), udon noodles and prepared sauce. Cook for 1-2 min, tossing everything together until noodles are well coated. For slightly crisped noodles, mix well then let them sit without stirring for 1-2 minutes to brown on the bottom.
Serve: Serve the yaki udon immediately. Garnish with extra green onions, toasted sesame seeds and chili flakes, if desired.