Add the broccoli florets, white beans, diced onion, and minced garlic to the slow cooker.
Pour the vegetable broth over the ingredients and stir once to combine.
Cover and cook on low for 6 hours until the broccoli is very tender.
Stir in the half-and-half, shredded cheddar cheese, salt, black pepper, and paprika.
Cover again and cook on low for 30 more minutes until the cheese melts fully.
Use a potato masher to lightly crush some of the white beans and broccoli for a thicker texture.