Melt the butter in a dutch oven or large pot.
Prepare and chop up your ingredients so that they are ready to add when you need them. Dice the potatoes into small pieces (see the post images for size), dice the carrots, dice the onion, and mince the garlic. Chop the broccoli florets into small pieces.
Add the garlic and onion to the dutch oven and cook them over medium low heat for 7-10 minutes, or until the onion is softened and translucent.
Add the carrots and cook them for 5 minutes over medium heat.
Add the potatoes and vegetable broth and bring it to a boil. Boil until the potatoes are tender and easily pierced with a fork.
Mix together the milk and flour in a bowl, then add it to the vegetable soup, stirring to combine.
Add in the finely diced broccoli and simmer for 10-15 minutes, stirring frequently.
Add the cheese and stir until melted and the soup thickens. The soup will continue to thicken the longer that it cooks, and the potatoes will start to fall apart and become part of the soup. Continue to simmer the soup to your desired thickness.
Salt to taste before serving.