Mix cranberry sauce, mayonnaise and ginger in small bowl. Set aside. Spread butter on 1 side of each slice of bread. Place 4 of the bread slices, buttered-side down, on cutting board. Top with turkey, cranberry spread, arugula leaves and cheese. Cover each with second slice of bread, buttered-side up.
Heat large skillet on medium heat. Place 2 of the sandwiches in skillet. Cook 2 to 4 minutes per side or until bread is toasted and cheese is melted. Repeat with remaining 2 sandwiches.