Chop the onion and mince the garlic, then prepare the broccoli florets, keeping all ingredients ready for cooking.
Add the onion, garlic, butter, salt, and pepper into the slow cooker. Turn on and cook on high until the onion becomes translucent and fragrant, about 5-6 minutes, stirring occasionally.
Pour in the vegetable broth, then add the broccoli florets. Cover the slow cooker and cook on low for 3 to 4 hours until the broccoli is tender and easily pierced with a fork.
Use an immersion blender directly in the slow cooker to blend the soup until smooth and creamy, or transfer in batches to a blender, then return to the slow cooker.
Stir in the shredded cheddar cheese and milk or cream, then cook on low for another 15-20 minutes until the cheese is melted and the soup is heated through. Taste and adjust with salt and pepper as needed.
Spoon the hot, velvety soup into bowls, garnish with extra cheese if desired, and serve immediately for a comforting, creamy meal.