Melt 2 tablespoons butter in medium saucepan. Add flour and cook 2 minutes. Whisk in milk. Heat to simmer. Cook over medium heat until thickened, stirring constantly. Remove from heat. Stir in cheese, salt, pepper and basil; set aside.
Heat remaining butter and oil in large skillet. Add chicken and stir over medium heat until cooked through. Stir into cheese sauce.
Cook onions and chopped vegetables in same skillet 5 minutes or until tender but still crunchy. Stir into cheese sauce. Pour over cooked pasta and serve with additional Parmesan cheese.