Cut the ends and the dark green leaves of the leeks off and discard, cut he leeks in half lengthwise. Place in a bowl of cold water for 10 minutes, then run under the faucet to remove any sand that may be lingering in between the layers. Peel off thick outer layers and discard.
Heat the olive oil over medium heat in a wide, heavy skillet that will accommodate all of the leeks in one layer. Place the leeks in the pan, cut side down, and cook, shaking the pan and moving them around with tongs, until they are lightly browned, 3 to 4 minutes.
Using tongs, turn the leeks over and cook on the other side until they are lightly browned, 3 to 4 minutes. Season with salt and pepper.
Turn the leeks back over so that the cut side is down. Peel off the outer layers if they are papery, as they will not soften when the leeks are braised.
Pour in the wine and stir to deglaze the bottom of the pan, then add enough water or stock to come just to the top of the leeks. Bring to a boil, reduce the heat and simmer uncovered for 20 to 25 minutes, until the leeks are thoroughly tender when pierced with a knife. Most of the liquid should have evaporated by this time. Meanwhile, preheat the broiler.
Using tongs, turn the leeks so that the flat side is up. If there is still a lot of liquid in the pan, pour it off. Sprinkle the Parmesan over the leeks. Place under the hot broiler until the cheese has melted and is beginning to color. Remove from the heat and serve