Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Whisk the flour, oats, cinnamon, baking powder, baking soda, salt and allspice in a medium bowl. Combine the butter, brown sugar, granulated sugar and eggs in a separate bowl; whisk until smooth. Stir in the grated apple and vanilla, then stir in the flour mixture until just combined.
Scoop heaping tablespoonfuls of the dough onto the prepared baking sheets, about 1 inch apart. Bake until just set around the edges, 10 to 12 minutes, switching the baking sheets halfway through. Let cool 5 minutes on the baking sheets, then transfer to a rack to cool completely.
Make the glaze: Whisk the confectioners' sugar and milk in a bowl; drizzle on the cookies.
Photograph by Andrew Purcell
Recipe courtesy of Food Network Magazine