Creamy Potato-Broccoli Soup

Combine chicken broth, chopped onion, and diced potatoes in a large pan. Cover, bring to a boil, reduce heat to a simmer, and cook for 20–25 minutes, or until potatoes are very tender when pierced with a fork. Drain off liquid and reserve. Remove 2 cups of potato and onion mixture and set aside. In the pan, mash the remaining potato mixture with an electric mixer. Mix in cooking liquid. Whisk in half-and-half, butter-flavor granules, salt, and pepper. Break apart broccoli. Stir broccoli and reserved potatoes and onions into soup. Return to stove, cover, and cook at a gentle simmer for 10–15 minutes, or until soup is heated through and broccoli is tender; stir periodically to prevent sticking. Spoon 1 cup into each bowl and sprinkle with cheese and chopped green onions.