Meaty portobellos take on a smoky edge when cooked over charcoal. Served with melted cheese, a fringe of peppery greens, and a slab of grilled onion on a high-quality roll, they make a great gourmet vegetarian stand-in for hamburgers. For the best flavor, make sure you allow at least an hour to marinate the mushrooms before grilling.
Place the mushrooms and onions on a baking sheet. Whisk together the olive oil, vinegar, thyme, salt, and pepper. Brush the marinade liberally onto the vegetables, coating all surfaces. Brush with more olive oil if needed. Let sit at room temperature for at least 1 hour.
Heat a barbecue grill to medium-high heat. Arrange the onions and the mushrooms, gill sides up, on the grill. Cook for 4 or 5 minutes, or until grill marks are noticeable. Turn the vegetables and move them to a cooler part of the grill. Cover the barbecue and cook until the mushrooms are very tender, 6 to 8 minutes. The onions should be crisp-tender but still retain their shape.
Flip the mushrooms so they're once again gill-side-up. Top each mushroom with a slice of cheese. Cover the grill and cook until the cheese melts, about 1 minute.
Arrange the bottom halves of the rolls on a platter. Spread with mayonnaise, if using. Top each half with some of the arugula, then with a mushroom, and a slice of onion. Spread the top halves of the rolls with mayonnaise, if using, and press onto the sandwiches. Serve hot.