In a skillet, brown chicken breasts in oil on one side. On each unbrowned side, place 1 piece of Swiss cheese and 1 to 2 spears of broccoli. Roll up (secure with a toothpick if desired); set aside.
In a saucepan, saute onion in butter until tender. Add flour and stir to make a smooth paste. Gradually add broth and milk; stir until thick. Add basil, celery salt and pepper.
Place half of the sauce in the bottom of greased 11x7-in. baking dish. Place chicken rolls over sauce. Cover with remaining sauce. Cover and bake at 350° for 20 minutes. Uncover and bake 10 minutes more or until chicken is no longer pink.