Cook noodles according to package directions; drain.
Combine broth and next 5 ingredients (broth through crushed red pepper) in a bowl; stir well with a whisk.
Heat 1 teaspoon oil in a large nonstick skillet over medium-high heat. Add the chicken; sauté 4 minutes. Add bell pepper; sauté 3 minutes. Remove from heat. Combine chicken mixture and noodles in a large bowl.
Heat 1 teaspoon oil in pan over medium heat. Add ginger and garlic; cook 15 seconds. Stir in broth mixture; cook 30 seconds, stirring constantly. Add the broth mixture and 1/4 cup green onions to noodle mixture; toss well. Sprinkle with 1/4 cup green onions.