Combine all sauce ingredients in a small bowl and set aside.
Cook udon noodles according to package directions, drain well, and set aside.
While noodles are cooking heat oil in a large pan over medium-high heat. Season chicken with salt and pepper to taste and add to the pan. Cook stirring occasionally until browned and no pink remains, about 5 minutes. Remove the chicken from the pan and set aside in a bowl.
Add the onions to the pan adding more oil if needed. Cook 2 minutes. Add remaining vegetables. Cook, stirring occasionally, just until tender-crisp, about 5 minutes.
Whisk the sauce (the cornstarch can settle to the bottom) and add to the vegetables. Bring to a simmer and let cook for about 1 minute.
Add the noodles and chicken with any juices to the pan and toss to combine with the sauce. Cook until heated through.
Garnish with sesame seeds and green onions if desired.
