Prepare leeks: Trim off the root ends of the leeks and split each leek lengthwise. Slice white and light green parts 1/4-inch-thick. Place sliced leeks in a bowl of cold water and swish the leeks around, separating layers, and letting any sand/dirt fall to the bottom. Scoop the leeks out (leaving the grit at the bottom) and drop into a colander to shake them off. It’s fine if they’re still damp.
Prepare potatoes: Peel potatoes and place potato peels in a bowl of cold water, so they don’t discolor while you make the soup. Slice potatoes 1/4-inch-thick.
Make the soup: Heat a medium-sized soup pot over medium-high heat and add butter. Once melted, add drained leeks, garlic, 1 teaspoon kosher salt, and many grinds of black pepper. Cook leeks, stirring occasionally, until softened but not browned, about 7 to 10 minutes.
Add the broth, sliced potatoes, and bay leaf and bring to a simmer. Cover the pot and simmer over medium-low heat until the potatoes are tender, about 15 minutes. You should easily be able to break the potatoes up with a spoon.
Make the crispy skins: While the soup simmers, drain potato peels and pat them dry. Line a baking sheet with parchment paper. Place peels on parchment and drizzle with olive oil, then sprinkle with salt. Toss to evenly coat then spread them out in a single layer. Bake peels for 10 to 15 minutes, until dry and crisp. Check in at the 10-minute mark and add more time only as needed. Once crisp, remove from the oven and set aside.
Finish the soup: Once potatoes are tender, remove and discard bay leaf, and use an immersion blender or transfer the soup to an upright blender and blend to desired consistency. My family doesn’t like fully pureed soups (they’re wrong, but…) so I only half-blend mine. Taste and add more seasoning as needed; I usually need at least another teaspoon of salt and much more pepper.
To serve: Ladle soup into bowls. Squeeze lemon juice over each, then dollop generously with sour cream, swirling it in. Shower each soup with chives, and sprinkle the top with some crispy skins, serving the rest on the side. Eat right away.
Do ahead: I keep the toppings separate when I store the leftover soup. It keeps in the fridge for up to 5 days.
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