Prepare the leeks by removing dark green tops and roots, slicing lengthwise to wash out grit, then cross-slicing into half-moons.
Crock Pot Method: Add leeks, potatoes, broth, garlic, thyme, salt, pepper, and butter to a 6-quart slow cooker. Cover and cook on Low for 6 to 8 hours or High for 3 to 4 hours until potatoes are very soft.
Instant Pot Method: Combine leeks, potatoes, broth, garlic, thyme, salt, pepper, and butter in the inner pot. Secure lid and set to Manual/Pressure Cook on High for 10 minutes. Perform a Quick Release once the cycle completes.
Once cooked, use an immersion blender directly in the pot to process the soup until completely smooth.
Stir in the heavy cream until fully incorporated and heated through.
Taste and adjust seasoning with additional salt or pepper as desired.
Serve hot garnished with chopped fresh chives.