In large saucepan or Dutch oven, cook rotini to desired doneness as directed on package, adding broccoli during last 5 minutes of cooking time.
Meanwhile, in medium saucepan, combine sour cream and flour; blend well. Stir in milk. Cook over medium heat, stirring constantly, until hot but not boiling. Stir in chicken, basil, mushrooms and 3 tablespoons of the cheese. Cook until thoroughly heated.
Drain rotini and broccoli; return to saucepan. Add chicken mixture; toss to coat. Place mixture in serving dish or on serving platter; sprinkle with remaining 1 tablespoon cheese.