Melt 2 tablespoons butter in a large pot over medium heat. Add ½ cup chopped onion and sauté until translucent.
Add 2 cups finely chopped carrot and 1 celery stick and cook over medium high heat until tender.
Add 2 tablespoons minced garlic and cook an additional 1-2 minutes.
Add 1 teaspoon salt, ½ teaspoon pepper, 4 large baking potatoes peeled and chopped, and 8 cups chicken broth. Bring to a boil and cook until the potatoes are slightly tender.
Mix 2 tablespoons flour with 1 cup water. Add to the soup and simmer until the soup is slightly thick to the desired consistency.
Add 1 cup milk and 4 cups chopped broccoli and simmer for 15 minutes or until the broccoli is just tender and the soup is heated through.
Stir in 4 cups shredded cheddar cheese and allow to fully melt.
Salt and pepper to taste.
