Heat up the grill to medium-high or turn on the oven broiler with the rack at the highest level. While the grill./broiler are heating, drizzle the oil and vinegar over the mushroom caps and rub all over with your hands. Season liberally with salt and pepper.
Put the mushrooms, cap-side-down, on the hot grill (set on a baking sheet and put under a broiler, if using). If grilling, cover the grill. Cook until they look quite tender on one side without burning, about 5 minutes. Turn over, cover (if grilling), and cook on the second side for 3 to 5 minutes until tender but not shriveled. For the last minute of cooking, set a slice of cheese on each mushroom to melt. Toast the buns on the grill or in the oven.
*You can find roasted bell peppers sold in jars in most markets or you can roast your own (a quick Google search will turn up plenty of instructions).