Cut off roots of the leeks and peel off any withered leaves. Cut off 2 inches from green tops. With a sharp knife slit the green parts in half lengthwise stopping an inch from root end. Make another slit, lengthwise and perpendicular. Spread leaves apart an rinse them under running water, carefully looking out for hidden grit. Cut leeks crosswise into 1 1/2-inch lengths.
Heat butter in a saucepan and add the leeks. Toss to combine them with the butter. Add the broth, cover and simmer for 15 to 20 minutes or until completely tender. You may do the recipe 2 to 3 days in advance up to this point. Over medium heat, evaporate the liquid from the leeks and when they are totally dry add heavy cream, starting with 1/4 cup or so and simmer, stirring continuously until the cream has been absorbed by the leeks. Add 1/4 cup more and repeat if you wish. Season to taste with salt and pepper.