01 - Place broccoli florets, diced onion, chopped carrots, celery, and minced garlic into a 4-quart or larger slow cooker.
02 - Pour the vegetable broth over the vegetables and stir to combine evenly.
03 - In a separate mixing bowl, whisk the melted butter and flour together to form a smooth paste. Add this mixture to the slow cooker and stir thoroughly to distribute.
04 - Cover the slow cooker and cook on LOW for 4 hours, or until all vegetables are fork-tender.
05 - Stir in the whole milk, heavy cream, cream cheese, salt, black pepper, smoked paprika, and nutmeg until well combined.
06 - Using an immersion blender, blend the soup directly in the slow cooker until smooth or to your preferred texture. Alternatively, carefully transfer half the soup to a countertop blender, puree, and return to the slow cooker.
07 - Stir in the shredded sharp cheddar and grated Parmesan cheeses. Cover and continue cooking on LOW for an additional 15–20 minutes, stirring occasionally, until the cheeses are fully melted and the soup is creamy.
08 - Taste the soup and adjust salt and pepper as needed. Ladle into bowls and garnish with extra shredded cheddar, fresh chives or parsley, and croutons. Serve hot.