Preheat oven to 350 degrees. Line baking sheets with parchment paper or baking mats.
Pour oats into a food processor and pulse about 10-12 times (you want it to be an even mix of ground and whole oats). In a medium size bowl, combine the oats, flour, cornstarch, baking powder, baking soda, salt, cinnamon and nutmeg, whisk together.
In a large mixing bowl or the bowl of a stand mixer, cream the butter, sugars and vanilla extract together until light and fluffy, about 2 minutes on high speed. With mixing speed on low, add the egg and mix until incorporated. Gradually add the dry ingredients, mixing just until combined. The dough is fairly soft so I prefer refrigerating it for 20-30 minutes before rolling, that way my hands aren't a mess and it gives it time to rest.
Using a standard size cookie scoop, place balls of dough onto each sheet. Place in the oven and bake for 10-14 minutes. I prefer crisp iced oatmeal cookies so I leave mine in for the full 14.
Remove from the oven and allow to cool on the sheets for 3 minutes before transferring to a wire rack to cool completely.
Once the cookies are cool, whisk together all of the ingredients for the glaze in a small mixing bowl. Dip the tops of the cookies in and swirl to remove any excess. Allow to set for 30 minutes.