1. Heat oven to 350 degrees F. Line 2 large baking sheets with nonstick foil. Grate apple into a bowl; add apple juice.
2. In a large bowl with a mixer, beat butter, sugar, egg and cinnamon until fluffy, about 2 minutes. With a spoon, stir in half the flour, the apple mixture, then remaining flour.
3. Drop 1/4-cupfuls of batter onto prepared sheets, 3 inches apart. Bake cookies at 350 degrees F for 17 minutes, until cookies are lightly browned. Remove to a rack; cool completely. Repeat.For Icing:
4. In medium-size bowl, blend confectioners sugar and apple juice until smooth. Transfer 1/3 cup icing to a small bowl; tint red. Transfer 3 tablespoons icing to a second bowl; tint green. Ice cookies on flat sides with white icing (about 1 tablespoon per cookie); let dry slightly. Pipe apples on top of cookies with red and green icing. Decorate same day as serving; store in single layer.