Begin by preparing your ingredients. Peel and dice the potatoes, chop the onion, and mince the garlic.
In the crockpot, combine the diced potatoes, broccoli florets, chopped onion, minced garlic, vegetable broth, salt, black pepper, and paprika.
Cut the butter into small pieces and add it to the crockpot. Stir all the ingredients together until well mixed.
Cover the crockpot and cook on low for 6 hours, or until the potatoes and broccoli are tender.
Once cooked, stir in the milk and shredded cheddar cheese. Mix well until the cheese is melted and the soup is creamy.
Use an immersion blender to blend the soup slightly, leaving some chunks for texture. If you don't have an immersion blender, carefully transfer half of the soup to a regular blender and blend until smooth, then return it to the crockpot.
Serve hot, garnished with extra cheddar cheese if desired.