01 - Melt the unsalted butter in a large pot over medium heat. Add diced onion, carrot, and celery; sauté for 4 to 5 minutes until softened.
02 - Stir in minced garlic and cook for one minute until fragrant.
03 - Sprinkle all-purpose flour over the vegetables and cook, stirring constantly, for 2 minutes to form a roux.
04 - Gradually whisk in vegetable broth to avoid lumps. Add broccoli florets and bring mixture to a simmer.
05 - Cover and cook for 10 to 12 minutes until broccoli is tender.
06 - Reduce heat to low. Using an immersion blender or standard blender in batches, purée soup to desired consistency, smooth or slightly chunky.
07 - Stir in whole milk, heavy cream, salt, black pepper, and nutmeg. Heat gently, avoiding boiling.
08 - Gradually add shredded cheddar cheese, stirring continuously until fully melted and soup attains creamy texture.
09 - Taste and adjust seasonings as needed. Serve hot.