Whisk together vinegar, oil, honey, mustard, pepper, and salt in a medium saucepan; bring to a simmer over medium, stirring occasionally to dissolve honey. Stir in shallots and garlic. Remove from heat.
Place apple slices in a large, shallow, heatproof dish; pour hot vinegar mixture over apples, and toss to coat. Let stand at room temperature 45 minutes, tossing occasionally. Drain apples, and discard vinegar mixture.
Spread a generous layer of butter on one side of each bread slice. Divide cheese, apple mixture, and arugula evenly on buttered sides of four bread slices. Top with remaining bread slices, butter sides down. Serve immediately.
