Chop onion, carrot, celery, garlic in a chopper
In a large soup pot, Melt butter.
Add chopped vegetables and saute on low heat until soft, about 5 minutes.
Add flour, salt and pepper to the pot and stir until smooth.
Add chicken broth, milk and potatoes and set heat to high until it comes to a boil, then cover and cook on low until potatoes are soft, about 10-15 minutes.
Add broccoli florets, parmesan cheese, and stir well. Adjust salt and pepper to taste. Cook uncovered until broccoli is cooked, about 5 minutes. Add cheddar cheese, stir well and remove from heat.