For posterity. And Aunt LoLo who has lost this recipe each of the four times I've sent it to her over the years. *mwah* I believe this originated with my mother-in-law's little sister.
Before you start, have cooked broccoli on hand, and 1 cup of broccoli cooking water. How much? Up to you. 1/2 a Costco bag is good for a big pot of this.
In a large pot, melt 1/4 cup butter and saute 2/3 cup chopped onion. (An onion. Chop an onion. Or two. I really like onions)
Add 1/2 cup flour and whisk well to make a white roux.
Add 2 cups chicken broth, 1 cup broccoli water, 4 cups dairy (could be half cream, I use all milk), 1/2 tsp Worchestershire sauce, 3/4 tsp salt (taste first!!). Mix all together, add broccoli and heat. Add 1 cup grated sharp cheddar cheese, or more to taste.
Don't let this boil, or it will curdle.