How to cook wild elk? very lean meat?

You want to grill or broil it carefully, and be careful not to overcook it. I would not cook it more than medium. Medium well will be already too dry. Another way is to stew or braise it for a long long time, with some added fat or oil. You can grind it up and use it like ground beef for hamburgers or meatloaf, too, but remember that you need to add extra fat. I like to grind up some bacon along with the elk for this reason.