Place the cubed potatoes, sliced leeks, vegetable broth, minced garlic, thyme, salt, and pepper into the slow cooker.
Stir once to combine everything evenly.
Cover and cook on low for 7 hours until the potatoes are very soft.
Use an immersion blender to puree the soup directly in the slow cooker until smooth.
Stir in the heavy cream until fully mixed.
Ladle into bowls and top each serving with chopped chives.