Easy Broccoli Cheese Soup

For the past few months, my ward's relief society newsletter started including recipes too.  How awesome is that!  This is one of the recipes included.  As soon as I saw it I knew it would make it on our dinner calendar.  I am so glad I made it because it's super easy, quick (less than 30 minutes!) and delicious!  I only made a few small changes to it; otherwise is just as it was in the newsletter.  :)  Enjoy!
-2 cups chicken broth
-1 small onion, finely diced
-2 cups milk
-4 Tbs salted butter
-1/3 cup flour
-1/2 tsp salt
-1/2 tsp pepper
-2 cups shredded cheddar cheese
-1/2 cup shredded Swiss cheese
-3 cups frozen chopped broccoli
Simmer broth and onion in a small saucepan for 15-20 minutes, covered.
Meanwhile, heat the milk in the microwave for a couple of minutes.
Melt the butter in a large saucepan.  Add the flour and create a roux.  Season with the salt and pepper.  Cook, stirring constantly for 2 minutes.  Gradually, whisk in the warm milk.  Stir well and cook for about 7 minutes or until the soup starts to thicken.
Add the broth and give it a good stir.  Add the cheeses and stir until completely melted.  Stir in the frozen broccoli.  Cook for about 5 minutes.  Serve immediately.  Enjoy!