01 - Melt butter in a large pot over medium heat. Add chopped onion and sauté until softened, about 3 to 4 minutes.
02 - Stir in minced garlic and cook for 1 minute until fragrant.
03 - Sprinkle flour over onions and garlic, stirring constantly for 2 minutes to form a roux.
04 - Gradually whisk in vegetable broth, ensuring a smooth base free of lumps.
05 - Add chopped broccoli and grated carrot. Bring to a gentle simmer, cover, and cook for 15 minutes until vegetables are tender.
06 - Reduce heat to low. Stir in whole milk and heavy cream, then simmer gently for 5 minutes without boiling.
07 - Using an immersion blender or traditional blender in batches, puree the soup to the desired consistency, smooth or slightly chunky.
08 - Gradually stir in shredded cheddar cheese, allowing it to fully melt. Season with salt, pepper, and optional nutmeg to taste.
09 - Ladle soup into bowls and garnish with additional cheddar and chopped chives or parsley if desired. Serve hot.