Trim the bulb end and dark green leaves from the leeks and remove the thick outer layers and discard. Cut each leek in half lengthwise.
Submerge the leeks in a large bowl of cold water for 10 minutes then run under a cold faucet to remove any sand and grit that may be stuck in between the layers.
Heat the oil over medium heat in a wide skillet that will accommodate all of the leeks in single 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, about 3 to 4 minutes.
Using tongs, carefully turn the leeks over and cook on the other side until they are lightly browned, or about another 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 the chicken stock, then add enough water to come just to the top of the leeks.
Bring to a boil then reduce the heat and simmer uncovered for 20 to 25 minutes, until the leeks are very tender. The majority of the braising liquid should have cooked off by this time.
Preheat the broiler and oil a baking dish large enough to hold the leeks if your skillet is not oven safe.
If your skillet is not oven safe transfer the leeks to the oiled baking dish using tongs with flat, cut side up. If your skillet is oven safe skip to step 9.
Pour off any remaining liquid in the pan and sprinkle the leeks with grated parmesan.
Place on the middle rack under the hot broiler until the cheese has melted and is beginning to brown. Remove from the oven, serve and enjoy.
