In a medium skillet with a lid, heat the oil over medium heat.
Add the leeks and the chicken broth; then cover and braise, stirring occasionally, for 12 minutes or until leeks are tender.
Remove the lid, and cook for about 10 minutes, or until the broth has almost evaporated.
Increase heat to high and sprinkle the sugar over the leeks.
Cook, stirring frequently, for another 10 minutes or until the leeks are lightly caramelized.