1. Trim the root end and the dark green tougher leaves from the leeks and discard those. Cut the remainder of the leeks vertically all the way along the stalk to expose the layers inside. Chop the halved leeks and transfer them to a sieve. Run a lot of cold tap water over them, tossing them frequently, to remove grit from inside the layers.
2. Melt the butter in a soup pot over medium heat. Add the leeks, the onion, and 1/2 teaspoon of the salt. Cook, stirring occasionally, until they’re softened, 5-6 minutes. Add the garlic and cook for 30 seconds.
3. Add the broth, the potatoes, and the remaining teaspoon of salt. Increase heat to high, cover, and bring to a boil. Reduce heat to a simmer and cook uncovered, stirring occasionally, until potatoes are fork tender, 8-10 minutes.
4. Remove pot from heat. Use an immersion blender to puree the soup, or puree it in batches using a standard blender and then return it to the soup pot. 
5. Add the ice cubes, the half and half, and the nutmeg.  Stir until ice cubes are melted. Transfer soup to the refrigerator to finish chilling. Serve cold topped with the chives.

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