While the mushrooms are cooking, prepare the grill for high direct heat. If you are using a gas grill, turn it on, cover it, and let it heat up for at least 10 minutes. (You want the grill to be very hot, which can difficult to do on a gas grill if the grill hasn't pre-heated long enough.) If you are using a charcoal grill, get the coals very hot. The grill is hot enough when you put your hand about an inch over the grill and you can only hold it there for 1 second. When the grill is hot, clean the grill grates with a wire scraper and then moisten a paper towel with vegetable oil. Using tongs, wipe down the grill grates with the oil-soaked towel.