Step 1Use the white, light and medium green parts of the leeks. Cut off the dark green tops (you can keep them in the refrigerator or freezer for flavoring soups and stocks).
Step 2Rinse the leeks briefly to remove surface dirt. Halve them lengthwise and rinse under cold water to remove any sand. Cut the leeks in thin slices. Put in a bowl of cold water and separate the pieces. Let them stand about 5 minutes, then lift them out of the water and drain in a colander. If the water is sandy, soak and drain them again.
Step 3Heat the oil in a heavy stew pan or deep saucepan. Add the leeks and salt and pepper to taste. Cover and cook over low heat, stirring occasionally, 5 minutes. Add the corn, thyme and 1/3 cup broth and bring to a simmer. Cover and cook, stirring from time to time, until the vegetables are tender, 15 minutes, adding more broth if the mixture becomes dry. If the leeks are soupy, uncover and cook, stirring often, to evaporate the excess liquid. Taste and adjust seasoning. Serve hot.