01 - In a large soup pot over medium heat, melt the butter. Add the finely chopped onion and grated carrot, sautéing for 4 to 5 minutes until they have softened.
02 - Stir in the minced garlic and cook for 1 minute more, until its aroma becomes fragrant.
03 - Sprinkle the all-purpose flour over the sautéed vegetables. Stir continuously for 2 minutes to cook out the raw flour taste, forming a light roux.
04 - Gradually whisk in the vegetable broth, ensuring the mixture is smooth and lump-free.
05 - Add the broccoli florets, salt, black pepper, and ground nutmeg (if using). Bring the mixture to a gentle simmer and cook for 10 to 12 minutes, or until the broccoli is tender.
06 - Reduce the heat to low. Stir in the whole milk and heavy cream, warming the soup gently without allowing it to boil.
07 - Using an immersion blender, carefully blend the soup to your preferred consistency, whether completely smooth or with some small vegetable chunks. Alternatively, transfer the soup in batches to a standard blender, process, and then return to the pot.
08 - Add the grated sharp cheddar cheese to the soup, a handful at a time. Stir continuously until the cheese is fully melted and the soup achieves a creamy texture.
09 - Taste the soup and adjust the seasoning with additional salt or pepper as needed.
10 - Serve the creamy broccoli cheddar soup hot, optionally garnished with extra grated cheddar cheese and fresh chopped chives or parsley.