01 - Melt butter in a large pot over medium heat. Add onion, carrot, and celery; sauté for 5 minutes until softened.
02 - Stir in garlic and cook for 1 minute. Add flour and cook, stirring constantly, for 2 minutes to form a roux.
03 - Gradually whisk in vegetable broth, ensuring no lumps remain. Add broccoli, thyme, nutmeg, salt, and pepper.
04 - Bring to a simmer, cover, and cook for 15–20 minutes, until broccoli is tender.
05 - Use an immersion blender to purée the soup until smooth, or transfer in batches to a stand blender.
06 - Reduce heat to low. Stir in milk and cream, then gradually add cheddar cheese, stirring until melted and smooth.
07 - Taste and adjust seasoning as needed. Serve hot, garnished with extra cheddar or fresh herbs if desired.