In a small saute pan, cook the chopped onion and garlic with the butter until softened and set aside.
Steam the broccoli in a steamer until tender, about 20-23 minutes or simmer on the stove in a saucepan with some water for 15-20 minutes. If using the saucepan method, drain when tender and return to saucepan.
Use a knife or back of a fork to break up the broccoli into small pieces, as large or small as you would like in the soup.
In a stockpot, pour in the chicken broth and add the carrots and hash browns. Simmer until ingredients are tender, 15-20 minutes
Add the onions from the sauce pan and the broccoli to the large stockpot and stir.
Stir the milk, and 3/4 of the cheddar cheese (about 9 oz.) and 6 oz. of the Velveeta into the soup until melted.