Place the broccoli florets, minced garlic, broth, and butter into the slow cooker.
Set the slow cooker to low and cook for about 4 hours until the broccoli is tender and the aroma is fragrant.
Use an immersion blender or transfer the mixture to a blender to puree until smooth and velvety.
Stir in the half-and-half (or heavy cream) and shredded cheddar cheese, blending until fully incorporated and the cheese melts into the soup.
Season with salt and pepper to taste, and give the soup a gentle stir to combine all flavors.
Serve hot, garnished with extra cheese or a sprinkle of fresh herbs if desired, enjoying its rich, smooth texture and bright color.