Place mushroom caps in a glass dish. Add dressing, turning mushrooms to coat. Marinate at room temperature for 15 minutes.
Heat gas grill or grill pan to medium high heat. Brush grill grate with oil. Grill buns to toast, if desired, about 1 1/2 minutes. Transfer mushrooms to grill, stemmed side up. Drizzle with any remaining dressing in dish. Grill, covered, 5 minutes.
Uncover mushrooms and turn over. Grill, covered, 4 minutes. Spoon 1 heaping tablespoon cheese onto each mushroom. Cover and grill 1 minute to melt cheese.
Place 1/4 cup baby arugula on each bun bottom. Top with one mushroom cap, some tomato slices and the top bun. Serve warm with onion rings or waffle cut fries.