Thin apple chips and crystallized ginger garnish a rich chocolate-applesauce cake; cider glaze and confectioners' sugar top each slice.
Combine cider and 1 cup confectioners' sugar in a small saucepan. Set over medium heat, and bring to a boil. Skim surface, removing any cider sediment that rises to the top. Reduce heat; simmer until liquid begins to thicken into a glaze, about 30 minutes.
Heat oven to 200 degrees. Squeeze juice from lemons into a small bowl filled with cold water. Using a very sharp knife, slice the apples into rounds as thinly as possible. Place rounds in the acidulated water. Line a baking sheet with a Silpat baking mat; set sheet aside.
Remove cider glaze from heat, and let stand until cool enough to touch. Carefully dip each apple round into the glaze, and place on the baking sheet.
Bake the apple rounds until they turn golden and the edges have curled, about 2 hours. Transfer apples to a wire rack, and let cool until crisp. Brush with the cider glaze.
Return cider glaze to heat. Slice ginger into 1/8-inch-thick slices crosswise, and add to cider. Cook over low heat until ginger is soft and translucent, about 30 minutes. Using a slotted spoon, transfer ginger to wire rack to cool. The liquid should have reduced to be just less than 3/4 cup.
Return glaze to heat; whisk in remaining 3 cups confectioners' sugar. Remove from heat. To test, pour a small amount over top of cake. If glaze is too thin, let cool slightly. If it is too thick, add a little more cider. Use when desired consistency is achieved.