In a large soup pot or Dutch oven, over low to medium heat, add the butter, onion, carrot, and celery. Sauté the vegetables in the butter for about 5 minutes.
Next, to the pan, add the flour, salt, pepper, and garlic salt. Cook stirring constantly for about 1-2 minutes.
At this point add the chicken broth* and potatoes. Simmer this over low heat for 5 minutes, then add the broccoli.
Continue to simmer the soup for 5-10 minutes more until the potatoes just start to get tender.
NOTE: I try not to cook the soup for too long or you will start seeing your potatoes dissolve into the soup.
At this point you can add the milk and cheese. Heat up over low heat until warmed thoroughly and then serve.