01 - Place the broccoli florets, diced onion, chopped carrots, diced celery, and minced garlic into a 4-quart or larger slow cooker.
02 - Pour the vegetable broth over the vegetables and stir to ensure everything is evenly combined.
03 - In a small mixing bowl, whisk together the melted butter and flour until a smooth paste forms. Add this roux to the slow cooker and stir thoroughly to incorporate.
04 - Cover the slow cooker and cook on the LOW setting 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 consistency. Alternatively, carefully transfer half of the soup to a countertop blender, blend until smooth, and return it to the slow cooker.
07 - Add the shredded sharp cheddar and grated Parmesan cheeses. Cover and continue cooking on LOW for an additional 15 to 20 minutes, stirring occasionally, until all the cheese is fully melted and the soup is creamy.
08 - Taste the soup and adjust the salt, pepper, or other seasonings as needed to suit your preference.
09 - Ladle the hot soup into bowls and garnish with extra shredded cheddar, fresh chives or parsley, and croutons as desired. Serve immediately.