In a crockpot, combine the chopped potatoes, broccoli, onion, garlic, vegetable broth, salt, black pepper, paprika, and cayenne pepper. Stir to combine.
Cover and cook on low for 6-8 hours or on high for 3-4 hours, until the potatoes are tender.
Once the cooking time is complete, use an immersion blender to puree the soup to your desired consistency. If you prefer a chunkier soup, blend only half of it.
Stir in the heavy cream and shredded cheddar cheese until melted and well combined. Adjust seasoning if necessary.
Serve hot, garnished with additional cheese or fresh herbs if desired.