Step 1

Melt butter in a large heavy pot over medium heat. Add leeks, onion, and garlic and season with salt. Cook, stirring often, until vegetables begin to soften, 5–7 minutes. Add 1 cup water and cook, stirring occasionally, until vegetables are very soft, 25–30 minutes. Let cool slightly, then transfer half of leek mixture to a blender and add half of milk, cream, and yogurt. Purée until very smooth; pour into a large bowl. Repeat with remaining leek mixture, milk, cream, and yogurt.

Step 2

Return soup to pot and heat over medium, adding water by ¼-cupfuls, until soup is thick enough to hold potatoes without sinking but thin enough that a spoon dragged across the top doesn’t leave a trail. Season with salt and pepper.

Step 3

DO AHEAD: Soup can be made 2 days ahead. Cover and chill.

Step 4

Rinse potato in a colander under cold water until water runs clear; pat completely dry.

Step 5

Fit a medium saucepan with thermometer; pour in oil to measure 2”. Heat over medium-high until thermometer registers 350°. Working in batches and returning oil to 350° between batches, add potato by the handful and fry, turning occasionally, until golden and crisp, about 2 minutes. Using a spider or slotted spoon, transfer potato to paper towels to drain. Fry parsley and rosemary until parsley is translucent, about 15 seconds. Add to potato and let cool; season with kosher salt. Serve soup topped with a nest of potatoes and herbs and seasoned with sea salt.

Step 6

DO AHEAD: Potatoes and herbs can be fried 2 hours ahead. Store uncovered at room temperature.