Melt the butter over medium heat in a large pot or Dutch oven.
Add the sliced leeks and cook for 8 to 10 minutes, stirring often, until softened but not browned.
Stir in the minced garlic and cook for 30 seconds until fragrant.
Add the diced potatoes, chicken broth, salt, and pepper to the pot.
Bring the mixture to a gentle boil, then reduce heat to low and simmer uncovered for 15 to 20 minutes, until the potatoes are very tender.
Remove the pot from heat and blend the soup until completely smooth using an immersion blender, or carefully transfer to a blender in batches.
Return the soup to low heat and stir in the heavy cream.
Taste and adjust seasoning with additional salt and pepper as needed.
Warm gently for 2 to 3 minutes without boiling.
Serve hot, garnished with chopped chives.