Preheat oven to 350° degrees F. Line 2 baking sheets with parchment paper; set aside. In a small bowl, toss together the shredded apple with the lemon juice and set aside.
For the cookies: In the bowl of a stand mixer, cream together butter and brown sugar with the paddle attachment until creamy, about 30 seconds. Add in the egg and vanilla and mix well until blended. Meanwhile, in a medium bowl, whisk together the flour, baking soda, salt, ground cinnamon, and ground nutmeg until fully combined.
Add half of the flour mixture into the beaten butter mixture and mix well, scraping down the bottom and sides of the bowl as needed. Add in the apple juice concentrate, mixing well. Lastly, add in the remaining flour mixture and mix until fully combined. Toss the shredded apple and juices and fold to combine.
Drop heaping 2-Tablespoon-size balls of dough onto the baking sheets about 2-3" apart from one another, using a wet finger to pat down any apple pieces sticking out. Bake for 10-12 minutes, rotating pans halfway through baking time to ensure even cooking. Cool on the baking sheets for about 10 minutes, then carefully transfer to wire racks to cool completely. Cookies are very soft and delicate.
For the Frosting: In a medium saucepan, combine the brown sugar, butter, 3 Tbsp apple juice concentrate and salt over medium heat and cook until butter melts and brown sugar dissolves, stirring occasionally to prevent it from scorching. Remove from heat and whisk in the confectioners' sugar until a thick icing comes together. Working quickly, ice one cookie at a time using an offset spatula or a knife and immediately garnish with chopped peanuts or pecans, if using. If the frosting hardens, thin it out over low heat with some more reserved apple juice concentrate, whisking it in to combine. I recommend keeping the icing over low heat and whisking it occasionally to keep it moistened and warm.
Store cookies at room temperature in an airtight container.