Preheat oven to 350 degrees. Pour oats into a food processor and pulse until partially ground, about 15 seconds. In a mixing bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, nutmeg and ground oats, for 30 seconds.
In the bowl of an electric stand mixer, fitted with the paddle attachment, whip butter, granulated sugar and light-brown sugar on medium-high speed until pale and fluffy, about 3 - 4 minutes. Add in eggs one at a time, mixing until combined after each addition. Stir in vanilla. With mixer set on low speed, slowly add in dry ingredients and mix just until combined (scrape bottom and sides of bowl as needed). Allow cookie dough to rest 10 minutes at room temperature.
Scoop dough out about 2 Tbsp at a time and drop onto a silpat or parchment paper lined baking sheet. Bake in preheated oven 11 - 15 minutes. Allow cookies to rest on baking sheet several minutes before transferring to a wire rack to cool completely.
In a mixing bowl whisk together powdered sugar and milk and spread icing over cookies. Return to wire rack and allow icing to set. Store in an airtight container at room temperature.