Heat the cider in a medium skillet over medium-high heat until it comes to a boil. Continue to cook until syrupy and reduced to about 2 tablespoons, 12 to 14 minutes. Set aside to cool slightly.
Whisk the flour, cream of tartar, baking soda, cinnamon and salt in a medium bowl until smooth. Beat the butter with 1/2 cup of the granulated sugar and the light brown sugar in a separate bowl on medium-high speed until fluffy and smooth, 2 to 3 minutes. Beat in the reduced cider and the egg (the mixture may look slightly curdled). Stir in the flour mixture and 1/4 cup of the chopped apple chips until combined.
Stir together the remaining 3 tablespoons of granulated sugar, 2 tablespoons of chopped apple chips and the apple pie spice in a small bowl. Roll heaping tablespoons of dough into balls, using lightly moistened hands if the dough is too sticky, and then roll in the spiced sugar mixture. Place the dough balls 3 inches apart on ungreased baking sheets and top with a few pieces of whole apple chips. Bake until the edges are just set but centers are still soft, 11 to 13 minutes, rotating pans halfway through the baking time. Cool 2 to 3 minutes on the baking sheet, and then transfer to a wire rack to cool completely.