01 - Place the broccoli florets, chopped onion, diced carrots, celery, and minced garlic into the slow cooker. Pour in the vegetable broth and sprinkle with dried thyme, stirring gently to combine.
02 - Cover the slow cooker and cook on the LOW setting for 4 hours, or until all vegetables are fork-tender.
03 - In a small saucepan over medium heat, melt the unsalted butter. Whisk in the all-purpose flour and cook for 1 to 2 minutes, stirring constantly, until a smooth paste forms. Gradually pour in the whole milk while whisking continuously until the mixture is smooth and slightly thickened.
04 - Pour the thickened milk mixture into the slow cooker. Add the softened cream cheese, shredded sharp cheddar, and grated Parmesan. Stir thoroughly until the cheeses begin to melt and integrate into the soup.
05 - Using an immersion blender directly in the slow cooker, or by transferring batches to a countertop blender, puree the soup to your preferred texture. Leave some broccoli pieces intact for a chunky consistency, or blend until completely smooth.
06 - Season the soup with salt, black pepper, and a pinch of cayenne pepper if desired. Continue heating on LOW for an additional 10 minutes, stirring occasionally, until the soup is uniformly creamy and heated through.
07 - Ladle the hot soup into bowls. Garnish with extra shredded cheddar cheese or croutons if desired, and serve immediately.