Prepare the vegetables by finely slicing the cabbage, cutting the carrot into thin matchsticks. Remove the stems from large shiitake mushrooms as they tend to get a bit woodsy and trim the stem for the small ones, then cut the mushrooms into bite-sized pieces or leave the small ones whole. Finally, cut the chicken breasts into bite-sized pieces and season with salt and pepper.
Cook the udon noodles according to package instructions if using dry noodles. Rinse after cooking. If using pre-cooked noodles, skip this step.
Meanwhile in a wok or a large pan, heat the oil and cook the chicken until sealed on the outside, then add the vegetables together with the sliced white parts of the green onions and continue stir frying for 5-7 minutes until cooked but still firm to the bite. The vegetables will reduce in volume by a half.
Add the udon noodles and the green onions to the pan, then add in the soy sauce, mirin and the dashi stock, toss to combine and heat through for 2-3 minutes. Drizzle with sesame oil before serving.