Place chopped potatoes and leeks in the slow cooker and set aside.
Heat olive oil in a skillet and add carrots and celery; cook for 4 minutes.
Stir in garlic and cook for 20 seconds.
Add carrot mixture to the slow cooker.
Stir in vegetable broth; add butter and thyme sprigs.
Close with the lid and cook on LOW for 6 to 8 hours, or until potatoes are very tender. You can also cook it on HIGH for 4 to 5 hours.
When done, remove the thyme sprigs and, if using, stir in the heavy cream.
Blend the soup with an immersion blender. If you only have a traditional blender, blend it in batches.
Taste for seasonings and adjust accordingly.
If soup is too thick, add water or vegetable broth to thin it out.
Ladle soup into bowls.
Garnish with parsley and fresh thyme leaves.