01 - Place the 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 together the melted butter and flour until a smooth paste forms. Add this mixture to the slow cooker and stir thoroughly to incorporate.
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, softened cream cheese, salt, black pepper, smoked paprika, and nutmeg until well combined.
06 - Using an immersion blender, purée the soup directly in the slow cooker until smooth or to your preferred texture. Alternatively, carefully transfer half of the soup to a countertop blender, blend until smooth, and return it 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 to 20 minutes, stirring occasionally, until the cheeses are fully melted and the soup is creamy.
08 - Taste the soup and adjust salt, pepper, or other seasonings as needed.
09 - Ladle the hot soup into bowls. Garnish with extra shredded cheddar, fresh chopped chives or parsley, and croutons as desired. Serve immediately.