01 - In a large pot, melt butter over medium heat. Add onion, carrot, and celery; sauté for 5 minutes until softened. Add garlic and cook for another minute until fragrant.
02 - Stir in the flour and cook for 1-2 minutes, stirring constantly, to form a smooth roux that will thicken the soup base.
03 - Gradually whisk in the vegetable broth, ensuring no lumps remain. Add broccoli florets and bring to a simmer. Cook for 10-12 minutes, until broccoli is tender.
04 - Reduce heat to low. Using an immersion blender (or in batches with a standard blender), blend the soup to your preferred texture—either completely velvety smooth or with some tender pieces for body.
05 - Stir in milk and heavy cream. Add grated cheddar cheese a handful at a time, stirring until fully melted and incorporated. Season with salt, pepper, and nutmeg to taste.
06 - Warm the bread in a 375°F oven or toaster until crispy and golden on the outside, about 5-7 minutes.
07 - Ladle the hot soup into warmed bowls and serve immediately with crusty bread on the side for dipping.