Cut away and discard the dark green leaves and roots from the leeks, leaving only the white and palest green sections. Cut them lengthwise and rinse in water to remove any dirt. Chop the leeks into pieces about ¼-inch wide.
Melt the butter over medium heat in a large Dutch oven, and add the leeks, onions, and shallots. Season generously with salt and pepper. Cook over medium heat for about 3-5 minutes. They should wilt but not brown. Meanwhile, cut the potato lengthwise into quarters, then cut crosswise into strips about ¼-inch thick.
Add the garlic and cook for about another minute, and then add the potatoes and sachet to cook for another 2 to 4 minutes.
Add the 5 ½ cups chicken broth to the mixture. Stir and taste, adjusting seasonings as needed. Increase the heat to medium-high and bring to a simmer. Reduce the heat and simmer for 30 minutes or until the potatoes are tender.
Remove the soup from the heat and let it stand and cool for 15 minutes. Remove the sachet. If serving chilled, prepare an ice bath.
Using an immersion blender, purée the soup until smooth. (Alternatively, you can use a blender, but be careful not to overfill it, as the hot liquid can spurt out the top.)
If using a blender, return the soup to the pan and bring to a simmer. Add the heavy cream and simmer for 5 minutes. If serving hot, stir in the chives. If refrigerating the soup, pour it into a container and place in an ice bath to cool. (Once refrigerated, it may thicken, so thin to desired consistency with more stock before serving). Enjoy!