01 - In a large soup pot set over medium heat, melt the butter. Add the finely chopped onion and grated carrot, sautéing for 4 to 5 minutes until they begin to soften.
02 - Stir in the minced garlic and cook for an additional minute until its aroma develops.
03 - Sprinkle the all-purpose flour over the sautéed vegetables. Stir continuously for 2 minutes to cook out the raw flour taste, forming a roux.
04 - Gradually whisk in the vegetable broth, ensuring the mixture remains smooth and free of lumps.
05 - Add the broccoli florets, salt, black pepper, and ground nutmeg (if using) to the pot. Bring the soup 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 mixture gently without allowing it to boil.
07 - Using an immersion blender, carefully blend the soup to your preferred consistency, whether entirely smooth or slightly chunky. Alternatively, transfer portions to a standard blender, process, and then return to the pot.
08 - Stir in the grated sharp cheddar cheese, adding it a handful at a time. Continue stirring until the cheese is completely melted and incorporated, yielding a creamy soup.
09 - Taste the soup and adjust seasonings as necessary to achieve optimal flavor.
10 - Serve the soup hot, optionally garnishing with extra grated cheddar cheese and a sprinkling of fresh chives or parsley.