Heat the butter over a medium heat until melted, add the leeks, and cook 5 minutes, stirring a few times. Add the broth, turn down the heat, cover, and simmer for 20 to 30 minutes until the leeks are tender, stirring now and again.

If there is a lot of liquid at the end, take the cover off and let most of it cook away. You want the leeks to be moist but not swimming in liquid. Put the leeks in a shallow baking dish, sprinkle the cheese over the leeks, and broil a few minutes.