Carefully wash and chop the leeks into rings after removing the green tops. Rinse thoroughly to remove dirt.

In a large pot, melt the butter over medium heat. Add the chopped leeks and onions, cook until softened, about 5 minutes.

Add the potatoes, chicken broth, salt, and pepper. Bring to a boil, then reduce heat and simmer for 20 minutes, or until potatoes are tender.

Use an immersion blender to blend the soup until smooth. Stir in the milk and heat through.

Serve hot, garnished with chopped chives or additional black pepper if desired.