Preheat your oven to 350 degrees and line two baking sheets with parchment paper or a silicone baking mat. Set aside.
In a food processor, add the rolled oats and pulse until coarse, about 10 seconds.
In a large bowl, add the ground oats, flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Whisk together.
In another large bowl (or the bowl of a stand mixer), add the butter and sugars and cream together with a hand mixer.
Add in the eggs, one at a time, mixing in between each addition. Add in the vanilla and mix.
Pour in the flour mixture, gradually, until well combined. The dough will be thick.
Use a cookie scoop to place 12 per baking sheet, evenly spaced.
Bake for 10-12 minutes until the edges begin to brown.
Cool on the baking sheet for about 5 minutes before transferring to a wire rack to cool completely.
Meanwhile, make the glaze in a shallow bowl. Add the powdered sugar and milk and whisk until smooth.
Take a cooled cookie and quickly dip the top into the glaze and let the excess drip off. I found it was easiest to dip straight down and hold it upside down while the extra drips off. Place back onto the wire rack until the glaze sets. I only dipped half of mine because I really liked the cookies plain, too.