Trim off about a half inch from the root end of the leek. Trim about 2 inches off the top of the leek. You primarily want to use the stalk as that will be most tender.
Slice the stalk into about 1/4 inch rounds. Rinse with water very well to remove any dirt or sand. Pat dry before cooking.
Add the butter, oil, and leeks to a nonstick pan over medium to medium low heat. Slowly cook the leeks until they become melt in your mouth tender, stirring occasionally, about 30 minutes or more if needed. Add a little more oil if needed throughout cooking. Season with salt and pepper a little a time throughout the cooking process.