Melt butter in a stock pot over medium-high heat. Cook and stir leeks and celery in butter until softened, 3 to 5 minutes. Pour chicken broth and vegetable broth into the pot; add potatoes, carrots, salt, and bay leaf.
Bring the mixture to a boil and cook until the potatoes and carrots are soft, about 20 minutes. Remove and discard bay leaf.
Blend soup with an immersion blender until smooth. Stir cream into the soup, reduce heat to medium-low, and simmer soup until thickened, about 20 minutes.
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