Place the butter in a large, deep pot over medium-high heat. When the butter is melted, add the shallots, potatoes, and leeks, sprinkle with salt and pepper, and cook until the veggies start to soften, 5-8 minutes.
Add the water and bring to a boil. Then, lower the heat so the soup bubbles gently and cook, stirring occasionally, until slightly thickened and vegetables become fully tender, about 20 minutes longer.
Carefully pour half of the soup into a blender and puree. When smooth, add the pureed soup back to the pan with the chunky remaining soup. Serve garnished with cilantro.