Preheat oven to 350 degrees. Line a cookie sheet with parchment paper.
In a large bowl beat the butter until smooth, about 45 seconds. Add the brown sugar, baking soda, apple pie spice and the salt. Beat on medium speed until well blended. Add the egg and beat just until combined. On low speed beat in the ½ cup apple juice. Beat in the flour until well blended. Stir in the apple.
Drop by rounded tablespoons 3 inches apart onto the cookie sheet. Bake for 8 to 10 minutes or until edges are light brown. The tops will look under baked. Allow to cool on the cookie sheet for 3 to 4 minutes before transferring to a wire rack to cool.
Over medium heat, cook and stir the ½ cup brown sugar, butter, and the apple juice until the sugar is dissolved and the butter is completely melted, remove from the heat. Whisk in the powdered sugar gradually, working quickly. Use the frosting immediately, if it begins to harden stir in hot water a teaspoon at a time until the desired consistency is reached.
Frost cooled cookies, sprinkle with pecans.
I roasted the pecans for 15 minutes in a 300 degree oven while mixing the cookie dough.
The frosting sets up very quickly, so I frosted 3 or 4 cookies at a time and then sprinkled with the pecans so they would stick.