Prepare barbecue (medium-high heat). Whisk vinaigrette and rosemary in small bowl. Brush cut sides of rolls with vinaigrette. Place rolls on grill, cut side down. Grill until lightly toasted, about 1 minute. Transfer to 2 plates.
Brush mushrooms on both sides with vinaigrette. Sprinkle generously with salt and pepper. Place mushrooms on grill, dark gill side down. Close grill or cover mushrooms with small metal roasting pan; cook until mushrooms begin to soften, brushing with vinaigrette once, about 5 minutes. Turn mushrooms over. Cover; grill until tender when pierced with knife, about 7 minutes longer. Place 2 cheese slices on each mushroom. Cover; grill until cheese melts, about 1 minute.
Place 1 mushroom on bottom half of each roll. Top each with tomato, arugula, and top half of roll.