In a medium size soup pot, melt 4 Tablespoons of unsalted butter. To the butter add, 1 chopped onion, 1 sliced carrot, and 1/4 teaspoon of salt and 1/4 teaspoon of black pepper. Sauté for 5 minutes, mixing occasionally.
Add in 3 Tablespoons of flour, cook for another minute and pour in 4 cups of vegetable broth. Bring to a boil.
To the pot add in 3 1/2 cups of cut broccoli heads. Cook for 10-15 minutes, or until broccoli becomes soft.
After broccoli has softened, pour in 1 cup of milk and add in 1 cup of shredded cheese. ( I used both mozzarella and cheddar)
Cook until the cheese completely melts, salt and pepper to taste. Enjoy!