In a 4-quart Dutch oven or pot, melt the butter over medium heat. Add the leeks, garlic and onion, and sweat until translucent (approximately) 10 to 12 minutes.
Add the potatoes and chicken stock, and bring to a simmer. Meanwhile, tie together the thyme, tarragon, oregano and bay leaf with kitchen twine, and add to the pot. Simmer until the potatoes are tender, 15 to 20 minutes.
Discard the herb bundle and, using an immersion blender, purée the soup until smooth.
Stir in the heavy cream, lime juice and zest, then season with salt and pepper. Let the soup cool to room temperature, then chill the soup completely.
When ready to serve, divide the soup between bowls and drizzle with a few drops of cream. Finish by garnishing with chives.