Based upon Joy of Cooking's recipe with a few minor adaptations. This recipe is gluten-free. It can be made dairy free by substituting margarine for the butter and either omitting the cream or substituting almond/cashew cream.
Prepare leeks by trimming off the tough, dark green leaves and the bottom of the root. Then cut length-wise and wash in between each leaf, as dirt deposits can gather deep inside the leek. Then, thinly slice.
Melt butter in a soup pot over low-medium heat.
Add sliced leeks, stirring occasionally. Cook until leeks are tender, but not browned. This may take 15-20 minutes.
Add sliced potatoes and chicken or vegetable stock. If using vegetable stock, you may add a little powdered chicken stock mix to enhance the flavor. Stir.
Bring mixture to a boil, then reduce heat and cook until potatoes are tender.
Once potatoes are tender, you can puree the mixture using a hand blender in the same pot or by doing batches in a blender. If you want a very fine texture, put the mixture through a sieve after pureeing.
Add cream, salt (optional), pepper, basil, and paprika. You can thicken the mixture by adding a tablespoon or two of flour or corn starch. You can thin the mixture by adding more water or stock. This depends upon your preferred texture and consistency.
Reheat gently to serve warm or hot. Or you can serve very cold with cut chives as garnish. Enjoy!