In a large skillet, heat oil over medium heat. Add mushrooms; cook until tender, stirring occasionally. Sprinkle with salt and pepper. Stir in balsamic vinegar. Cook, uncovered, until all of the vinegar has evaporated, stirring occasionally. If desired, cover and chill cooked mushrooms for up to 3 days.
Spread cooked mushrooms over browned side of a flatbread. Sprinkle with cheese before returning flatbread to the grill. Sprinkle with marjoram before serving. Cut into wedges. Makes 6 appetizer servings.