Pour oats into a food processor or high-powered blender and pulse until partially ground, about 15 seconds. In a large bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, nutmeg and ground oats until completely incorporated.
In a separate large bowl, cream together butter, sugar and brown sugar until light and fluffy (about 3 minutes). Add in eggs one at a time, mixing until combined after each addition. Stir in vanilla.
Slowly add in dry ingredients to wet ingredients and mix just until combined.
Scoop dough out onto baking sheet (I used a silpat liner) in heaping tablespoonfuls. Bake in preheated oven for 11 - 15 minutes. Allow cookies to rest on baking sheet several minutes before transferring to a wire rack to cool completely.
For the icing, whisk together powdered sugar and milk in a mixing bowl and dip tops of cooled cookies in icing and allow excess to run off or alternately spread icing over cookies. Return to wire rack and allow icing to set. Store in an airtight container at room temperature.