Add broccoli, onion, carrots, garlic, chicken broth, salt, pepper, and smoked paprika to your slow cooker. Give it a quick stir.
Cover and cook on LOW for 4 hours or HIGH for 2 hours, until the veggies are soft.
Use an immersion blender to puree the soup right in the pot, or carefully transfer half to a blender and blend until smooth (leave some chunks for texture if you like!).
Stir in the heavy cream and butter. If you want a thicker soup, mix cornstarch and water in a small bowl, then stir into the soup.
Add the cheddar cheese a handful at a time, stirring until melted and smooth.
Ladle into bowls, top with extra cheese or a sprinkle of black pepper, and enjoy!