While pasta is cooking prepare the cheese and milk mixture. In large pot on medium heat, add butter and onion. Cook onions till soft, then add flour. Stir flour quickly to combine the flour and onion rue mixture. Add milk and chicken broth.
Grease a large 9 x 12 baking dish.
Preheat oven to 375 degrees. In large pot, boil water and cook pasta to instructions, but leaving pasta a little al dente. About 3 minutes before draining pasta, add broccoli to cook in water alongside pasta. I like my broccoli a little firm, so that it doesn't end up mushy after the bake. You can decide how cooked you want your broccoli by adding it in sooner.