Halve the leeks lengthwise, then slice them into half-moon shapes approximately 1/2-inch thick. Rinse the leeks thoroughly to remove any dirt.
In a large pot, melt the butter over medium-high heat. Add the leeks and cook, stirring frequently, until they’re tender. Do not let brown.
Add the cauliflower and chicken broth. Turn heat to high and bring to boil. Reduce to simmer and cook until cauliflower is tender. Add the yogurt and return the soup almost to a boil. Remove from heat.
Purée the soup, in blender or food processor, as finely as possible. Season with salt and freshly ground pepper. Add milk to thin if needed, and return to pan and reheat over medium heat.