In a medium bowl, whisk together the flour, baking soda, salt, cinnamon, and nutmeg.
In a large bowl, using an electric mixer, beat the butter and sugars until light and fluffy, about 3-4 minutes. Add applesauce, egg, and vanilla and mix until combined. Scrape sides of the bowl as needed. The mixture may look a little curdle. Add the flour mixture and beat until just combined. Add shredded apples, oats, and pecans. Mix until combined. Scrape the sides of the bowl as needed.
Store the dough in an airtight container and place in the refrigerator for at least 2 hours.
Prepare the icing by mixing the brown sugar and milk until all the sugar is dissolved. Add powder sugar, cinnamon, and nutmeg and combine until smooth.
Preheat the air fryer to 350°F and set to 12 minutes. Line the air fryer rack with parchment paper. Drop 3 tablespoons of dough onto the rack; gently flatten the dough with your palm. Depending on the capacity of your air fryer, drop 2 - 3 more balls of dough, spacing them 1-inch apart. Bake until golden brown. Remove from air fryer and place on a cooling rack. Repeat the process with the rest of the dough.
Using a spoon, add icing to the cookies. Serve.