In a small saucepan, saute the onion in the butter until tender.
Next, stir in the flour, salt and pepper. Next, gradually add the milk, followed by the chicken broth. Bring the soup to a boil, then cook and stir until it has thickened, about two minutes. Editor's Tip: Let the flour cook for one to two minutes before adding the milk. This will help cook off any raw flour taste in the soup and create a smoother consistency.
Add the broccoli. Cook and stir until the broccoli is tender and the soup is heated through. Editor's Tip: Keep in mind that adding frozen broccoli to the soup will result in a thinner soup because it holds extra moisture. To maintain the soup's consistency, thaw and drain frozen broccoli before adding it to the soup.
Remove the soup from the heat. Stir in the cheese until it has melted. Serve the soup hot.
