Bring a large pot of lightly salted water to a boil. Add pasta and cook for 8 to 10 minutes or until al dente; drain. Place in large bowl with chicken.
In large skillet or wok, heat 1 tablespoon oil. Cook carrots with 1/4 teaspoon salt until tender-crisp, 3 to 5 minutes. Remove carrots, place in bowl with chicken and pasta. Add 2 tablespoons oil to wok and cook mushrooms with 1/4 teaspoon salt until tender, 5 minutes. Remove mushrooms and place in bowl.
Add 2 more tablespoons oil to wok, stir-fry broccoli, cauliflower and onion until coated with oil. Add 1/2 teaspoon salt and 1/4 cup water, cover and cook 5 to 10 minutes, until tender-crisp, stirring occasionally. Remove vegetables with slotted spoon and toss with pasta and chicken.
Mix in green onions, soy sauce and sesame oil, adjusting soy and sesame to taste. Serve warm or cold.