Chop onion, carrot, celery, garlic in a chopper or mini food processor.
In a large soup pot, melt butter. Add chopped vegetables and sauté 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 and American cheese, stir well and remove from heat.
Using an immersion blender, quickly blend part of the soup for a quick second or two. If you don't have an immersion blender, remove about 1-2 cups of potatoes and broccoli, place it in your chopper, then add it back to the soup. This helps thicken it a bit.