Trim off and discard the root ends of the leeks. Trim off and discard any dried out or browned ends of the leek greens. Cut the leeks in half lengthwise, slice them crosswise into bite-size pieces. You can use the dark green parts of the leek; chop it all up, just getting rid of any browned or dried ends.
Put the leeks in a large bowl and cover with cool water. Swish the leeks around to loosen any dirt or grit. If you noticed large clumps of grit when you were chopping the leeks, now is the time to make sure that gets rinsed off. Let the leeks sit a few minutes for the dirt and grit to settle to the bottom of the bowl.
In a large frying pan or sauté pan over medium-high heat, melt 1 tablespoon of the butter. Lift the leeks out of the water and add them to the pan. Do not drain the leeks into a colander, since that will dump the dirt back onto the leeks.
Sprinkle the leeks with the salt and stir to combine.
Cover and reduce the heat to medium-low. Cook, stirring every 5 minutes or so until the leeks are supremely tender, 30 to 40 minutes total.
Take the pan off the heat and add the remaining tablespoon of butter. Stir until the butter is melted into the leeks. Serve hot or warm.