01 - Melt butter in a large pot over medium heat. Add onion, carrot, celery, and garlic, cooking for 4 to 5 minutes until softened and fragrant.
02 - Stir in flour and cook, stirring constantly, for 1 to 2 minutes until lightly golden.
03 - Gradually whisk in vegetable broth to eliminate lumps. Add broccoli, salt, pepper, and nutmeg.
04 - Bring mixture to a simmer, cover, and cook for 15 to 18 minutes until broccoli is very tender.
05 - Use an immersion blender to purée the soup directly until smooth, leaving some texture if preferred. Alternatively, blend in batches in a countertop blender.
06 - Return soup to low heat. Stir in milk and heavy cream, heating gently for 2 minutes.
07 - Gradually add shredded cheddar cheese, stirring until melted and smooth. Adjust seasoning with salt, pepper, and cayenne as desired.
08 - Toast bread slices or warm the baguette in the oven.
09 - Ladle soup into bowls and serve hot alongside the bread.