Preparation Brush 1 side of each bread slice lightly with oil, using about 1 tablespoon total. On 2 slices, arrange half of cheese, then tomato, basil, and remaining cheese. Top with remaining bread slices, oil side down. Brush tops of sandwiches with about 1 tablespoon of remaining oil. Heat medium skillet or range-top grill pan over medium heat. Place sandwiches, oil side down, in skillet. Brush tops with remaining 1 tablespoon oil. Cook until sandwiches are golden brown on both sides and cheese is melted, turning once, about 5 minutes per side. Toss arugula with vinaigrette and divide between 2 plates. Cut sandwiches in half or quarters and arrange around arugula. Serve.
Preparation Brush 1 side of each bread slice lightly with oil, using about 1 tablespoon total. On 2 slices, arrange half of cheese, then tomato, basil, and remaining cheese. Top with remaining bread slices, oil side down. Brush tops of sandwiches with about 1 tablespoon of remaining oil. Heat medium skillet or range-top grill pan over medium heat. Place sandwiches, oil side down, in skillet. Brush tops with remaining 1 tablespoon oil. Cook until sandwiches are golden brown on both sides and cheese is melted, turning once, about 5 minutes per side. Toss arugula with vinaigrette and divide between 2 plates. Cut sandwiches in half or quarters and arrange around arugula. Serve.