Add the oats to a blender or food processor and pulse 5-6 times. Be careful not to over-blend the oats! You should have a mix of fine and larger pieces.
In a medium bowl, whisk together the pulsed oats, flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a large bowl, beat the butter, granulated sugar, and brown sugar together on medium-high speed until light and fluffy, about 4 minutes. Add eggs, molasses, and vanilla and mix until combined.
Add the dry ingredients to the wet and mix until just combined. Cover the bowl and chill in the refrigerator for an hour.
When the dough is nearly done chilling, preheat the oven to 350 degrees Fahrenheit and line two baking sheets with parchment paper.
Use a medium cookie scoop to scoop the dough onto the prepared baking sheets, leaving 2 inches between them. Bake for 10-12 minutes, until the edges are golden brown. Remove the cookies from the oven and let them cool on the pan for 3 minutes before transferring them to a cooling rack to finish cooling.
Once the cookies are completely cooled, make the glaze by whisking the milk into the powdered sugar. It should be fairly thick so it doesn't run into the ridges and crevices of the cookie. This is what gives it that signature iced oatmeal cookie look. If the icing is too runny, add a little more powdered sugar. It should stick but not run.
Carefully dip the very tops of the cookies into the icing and shake off the excess. Don't dip the cookies too deep, otherwise, they won't have the right look for a classic iced oatmeal cookie. Let the icing set for an hour or two before serving.