Step 1

Chop onion, carrot, celery, garlic in a chopper or mini food processor.

Step 2

In a large pot, melt the butter. Add the chopped vegetables, salt and pepper and sauté on low heat until soft, about 5 minutes.

Step 3

Stir in the flour to create a roux, cooking for 2 minutes to remove the raw flour taste.

Step 4

Gradually whisk in the chicken broth, ensuring there are no lumps.

Step 5

Add the milk and increase heat to high until it comes to a boil, then add broccoli florets, Parmesan cheese, and stir well.

Step 6

Adjust salt and pepper to taste. Cook uncovered over low heat until broccoli is cooked, about 10 minutes.

Step 7

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 soup and blend, then add it back to the soup. This helps thicken it a bit.

Step 8

Gently stir in the shredded cheddar cheese, allowing it to melt smoothly.

Step 9

Serve the soup hot, garnished with additional cheddar cheese, if desired.