In a large soup pot, melt the butter over medium heat. Add the chopped leeks and smashed garlic cloves, stirring frequently, until the leeks soften and become translucent, approximately 10 minutes. Ensure the heat remains moderate to avoid browning the leeks.

Incorporate the diced potatoes, broth, bay leaves, thyme sprigs, salt, and black pepper into the pot. Increase the heat to bring the mixture to a rolling boil, then cover and lower the heat to a gentle simmer. Let the soup cook for about 15 minutes, or until the potatoes are tender enough to break apart with a fork.

Remove and discard the bay leaves and thyme sprigs. Using an immersion blender, carefully blend the soup directly in the pot until smooth. (Alternatively, if using a standard blender, work in batches and blend until smooth—refer to note for details).

Pour in the heavy cream and bring the soup back to a gentle simmer. Taste and adjust seasoning with additional salt and pepper if needed. If the soup appears too thin, let it simmer further to thicken; if too thick, add a splash of broth or water to reach the desired consistency. Serve with a sprinkle of fresh chives for a vibrant finish.