Combine chicken broth, onion, potatoes, carrots and broccoli in a large stock pot and bring to a boil over high heat. Cook until the potatoes are fork tender (about 10minutes depending on how large the chunks are).

Add the half and half.

In a small bowl, melt the butter and add the flour, whisking until smooth. Add to the soup along with the cheese. Mix well and bring to a simmer for 5 minutes before serving.