Bring dark-green leek pieces, broth, and water to boil in large saucepan over high heat. Reduce heat to low, cover, and simmer 20 minutes. Strain broth through fine-mesh strainer into medium bowl, pressing on solids to extract as much liquid as possible; set aside. Discard solids in strainer and rinse out saucepan.
Melt butter in now-empty saucepan over medium-low heat. When butter foams, stir in sliced leeks, onion, and salt. Reduce heat to low and cook, stirring frequently, until vegetables are softened, about 10 minutes.
Increase heat to high, stir in reserved broth, potato, bay leaf, and herb sprig and bring to boil. Reduce heat to low and simmer until potatoes are tender, about 10 minutes. Add toasted bread and simmer until bread is completely saturated and starts to break down, about 5 minutes.
Remove and discard bay leaf and herb sprig. Transfer half of soup to blender and process until smooth and creamy, 2 to 3 minutes. Transfer to large bowl and repeat with remaining soup. Return soup to saucepan and bring to simmer over medium-low heat; season with salt and pepper to taste. Garnish and serve.