Preheat oven to 350 degrees F. Line a baking sheet with parchment paper and set aside. In a large bowl, combine the butter, brown sugar, egg, vanilla and apple juice (make sure the apple juice is room temp or warmer or it will cause the butter not to mix well). Mix until creamy.
Add the shredded apples and stir until combined. In a medium bowl, combine the flour, baking soda, salt and apple pie spice (or cinnamon). Mix well and gradually add to the batter, stirring until combined.
Drop large spoonfuls of batter onto the prepared baking sheet and bake in the preheated oven for 12-14 minutes. Let cookies cool, then prepare the frosting.
Combine the brown sugar, butter and apple juice in a medium saucepan. Cook over Medium heat until the sugar has dissolved. Remove from heat and stir in the powdered sugar. Stir until creamy and immediately spoon the frosting over the cookies, topping with a sprinkling of sea salt.