Heat the butter in a large Dutch oven over medium heat. Once the butter has melted, add the onions, carrots, and celery, then sauté for 3-4 minutes.
Add the minced garlic and cook, stirring constantly for 1 minute.
Whisk in 1 tablespoon of flour until well combined and cook for 2 minutes.
Slowly add the vegetable or chicken broth and mix well.
Add the potatoes, then cover with a lid and simmer for 35-40 minutes; add the broccoli florets and cook for 15-20 minutes or until all the vegetables are softened.
Add the last tablespoon of flour to the cheese and toss to coat evenly. Side Note: Tossing the cheese with the flour prevents the cheese from separating when it's added to the soup.
Add the cheese to the soup, along with the milk. Stir frequently until the cheese has melted and is well combined. Taste and re-season if needed.