2 Tablespoons butter
1/2 cup chopped onion
1 cup finely chopped carrot
1 celery stick, finely chopped (feel free to add more celery if you like celery; I don't, so I don't add much)
1 Tablespoon minced garlic
4 cups chicken broth (I usually use a mixture of low-sodium chicken broth and broth made with chicken soup base because it gives it such nice flavor)
2 large baking potatoes, peeled and chopped
1 Tablespoon flour
1/2 cup water
2/3 cup milk
2 cups chopped broccoli
2 heaping cups shredded cheddar cheese