Heat oven to 375°F.
Trim off the root ends of the leeks and split each leek lengthwise. Slice white and light green parts ¼-inch-thick. Place sliced leeks in a bowl of cold water and swish around, separating layers, and letting any sand/dirt fall to the bottom. Scoop the leeks out and drop into a colander to shake them off.
Peel potatoes and place potato peels in a bowl of cold water. Slice potatoes ¼-inch-thick.
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 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.
While the soup simmers, drain potato peels and pat them dry. Line a baking sheet with parchment paper. Place peels on parchment, drizzle with olive oil, and sprinkle with salt. Bake peels for 10 to 15 minutes until dry and crisp.
Once potatoes are tender, remove and discard bay leaf, and blend the soup to desired consistency. Taste and add more seasoning as needed.
Ladle soup into bowls. Squeeze lemon juice over each, dollop with sour cream, and sprinkle with chives and crispy skins. Serve immediately.
