Preheat the grill to medium-high heat.
Place the mushrooms in a baking dish or pie plate. Whisk together the vinegar, oil, salt and pepper until blended. Brush the mushrooms with the vinegar mixture; let stand for up to 15 minutes.
Place the mushrooms on the grill rack and grill until tender, 4 to 6 minutes per side. Transfer to a plate.
Meanwhile, combine the yogurt, rosemary and zest and juice of the lime in a small bowl until blended. Brush the bread slices with the yogurt mixture. Layer half the slices with the mushrooms, arugula, tomatoes, onions and red pepper strips. Top with the remaining bread slices. Serve immediately.