To prepare the leeks, cut off the ends and the dark green leaves. Halve the leeks lengthwise. Place in a large bowl of cold water for 10 minutes. Hold the leeks under the faucet to wash off any sand or dirt that may be between layers. Discard the thick outer layers.
Heat a large, heavy-bottomed skillet over medium heat and add the oil.
After the oil heats, place the leeks, cut-side down, in one layer.
Move the leeks around with tongs until lightly browned, about 3-4 minutes. Carefully, turn the leeks over and cook another 3-4 minutes or until lightly browned.
Season with salt and pepper and flip the leeks to cut-side down.
Remove the outer, papery layers (if any) as they won’t soften.
Add wine and stir to deglaze the pan. Add enough chicken stock to be level to the tops of the leeks.
Bring to a boil, reduce the heat to low, and simmer, uncovered about 15-20 minutes or until leeks are tender. Most of the liquid will have evaporated.
Add the butter and stir gently. Leave the pan on low and cook just long enough for any liquid to evaporate, about 1 minute. If the liquid hasn’t evaporated by the time the leeks have softened, pour off the excess liquid and keep refrigerated (up to 1 week) for another use.
Flip the leeks over to cut-side up and sprinkle with cheese.
If desired, set the oven to broil and make sure your pan is broiler-safe. Place the leeks under the broiler just long enough to brown.
Alternately, the leeks are just as delicious served without broiling - just give the cheese long enough to melt and serve.