In a large skillet over medium heat, melt the 4 tablespoons of unsalted butter. Add the thinly sliced leeks and cook, stirring occasionally, for about 8-10 minutes until they are very soft and translucent. Stir in the 4 cloves of minced garlic and cook for 1 more minute until fragrant.
Transfer the cooked leek and garlic mixture to the bowl of a 6-quart or larger slow cooker. Add the 3 pounds of peeled and cubed potatoes, 6 cups of low-sodium chicken broth, 1 teaspoon of kosher salt, and 1/2 teaspoon of black pepper, stirring to combine.
Cover the slow cooker and cook on the LOW setting for 7 hours, or on the HIGH setting for 4 hours, until the potatoes are extremely tender and easily pierced with a fork.
Carefully puree the soup directly in the slow cooker pot using an immersion blender until it is completely smooth and creamy. Alternatively, carefully transfer the soup in batches to a standing blender, cover with the lid and a towel to prevent splatters, and blend until smooth before returning it to the slow cooker.
Stir the 1 cup of room-temperature heavy cream into the blended soup until it is fully incorporated. Taste and season with additional kosher salt and black pepper if desired.
Ladle the hot soup into bowls, garnish each serving with a sprinkle of the 1/4 cup of chopped fresh chives, and serve immediately.
