Add everything EXCEPT the heavy cream and cheddar cheese to the slow cooker. Secure lid and cook on low for 5½ hours.
Remove lid and ladle ⅓ of the soup in to a bowl and blitz with an immersion blender (or into a blender, just use caution). Return to slow cooker and add heavy cream and cheese. Stir to combine. Secure lid and cook for remaining 30 minutes.
Serve with a sprinkle of additional cheese and dried chives on top.