In food processor, blend mayonnaise, basil, 1 tablespoon vinegar and garlic powder. In small bowl, whisk together remaining vinegar, the olive oil, salt and pepper.
Heat grill. Grill onion rolls, cut side down, until toasted, about 1-1/2 minutes. Brush mushrooms and tomato slices with balsamic-oil mixture. Grill mushrooms 8 to 10 minutes per side and tomatoes 2 minutes per side, brushing with additional balsamic-oil dressing, or until tender.
Spread generous 1 tablespoon basil-mayonnaise mixture on each cut side of onion rolls. Evenly divide arugula among 6 onion roll bottoms. Top with a portabella mushroom, a tomato slice and onion roll top. Serve immediately. Makes 6 servings.