Spread 1 side of each bread slice with 1/2 tbsp. butter. Spread the other side of 2 bread slices with 1/2 tbsp. apple butter each; top each with half the turkey, brie, and arugula, then a few apple slices. Spread the bare sides of remaining 2 slices of bread with remaining apple butter, then place on sandwiches, buttered side up.
Put 1 sandwich in a medium frying pan and cook, covered, over medium heat until golden brown on bottom, 3 to 4 minutes. Turn sandwich over and continue to cook until cheese is slightly melted and second side is golden brown, 3 to 4 more minutes. Repeat with remaining sandwich. Serve sandwiches with remaining apple slices.
Note: Nutritional analysis is per serving.