Preheat the oven to 350°F (175°C).
Brown the butter in a saucepan over medium heat and let it cool to room temperature.
In a medium bowl, whisk together flour, baking soda, salt, and apple pie spice.
In a separate mixing bowl, cream together the browned butter, brown sugar, and granulated sugar until creamy. Add the egg and vanilla extract, mixing until combined.
Gradually add the dry ingredients to the wet mixture and mix until well combined. Stir in rolled oats, grated Granny Smith apple.
Drop dough by rounded tablespoons onto parchment-lined baking sheets. Flatten slightly with the bottom of a glass. Bake for 10-12 minutes or until the edges are golden brown. Cool on a wire rack.
Caramel Icing: In a small bowl, whisk together powdered sugar, apple cider or juice, and caramel sauce until smooth. Drizzle the icing generously over the cooled cookies. Allow the glaze to set for about 1 hour at room temperature.