Melt the butter over medium heat in a large soup pot. Add the leeks and garlic, cooking while stirring regularly until soft and wilted, approximately 10 minutes. Adjust the heat as needed to avoid browning.

Stir in the chopped potatoes, broth, bay leaves, thyme sprigs, salt, and black pepper. Bring the mixture to a boil, then cover and reduce heat to low. Simmer gently for 15 minutes or until the potatoes are very soft.

Remove the thyme sprigs and bay leaves. Use a hand-held immersion blender to purée the soup until smooth. Alternatively, blend in batches in a standard blender, taking care to vent the lid to release heat.

Return the pureed soup to the pot if using a blender. Add the heavy cream and bring the soup to a gentle simmer. Taste and adjust seasoning with additional salt and pepper. If the soup is too thin, simmer uncovered to thicken; if too thick, add water or stock to reach desired consistency.

Ladle the soup into bowls and garnish with finely chopped chives. Serve hot.