Use a paper towel to clean the mushrooms as best as you can.
Heat a pan on medium high. Add oil and half of the vinegar, add the mushrooms cap side down. Cook one to two minutes before turning. Add the rest of the vinegar. Cook until they turn a golden-brown on the cap-side, turning a few more times if you've like. The vinegar-and-oil should be a murky brown color by the time they're done.
Place cheese on the cap and let it melt before placing on a bun with mayo and onion.
Enjoy!
Note: I don't usually measure. Adjust the measurements according to your tastes.