Preheat oven to 400 degrees. Grease a medium size casserole dish; set aside. Saute onion in butter over medium heat for 3 minutes. Mix in flour. Gradually stir in broth, combining well.
Gradually add milk to saucepan. Cook, stirring continually, until thickened, about 5 minutes. Stir in salt and pepper to taste.
Bring 2 quarts of water to a boil. Add noodles and cook for 6 minutes. Drain noodles well. Spread noodles in prepared baking dish. Place chicken and broccoli over noodles.
Pour sauce over chicken and broccoli. Sprinkle cheese over top. Bake for 20 minutes or until cheese melts.