Fit a large stand mixer with a paddle attachment, then beat the 1/2 cup butter, 1/4 cup granulated sugar and 1/3 cup brown sugar until creamy. Place the cracked 1 large egg and 1 tablespoon vanilla extract in the bowl and mix until combined.
Whisk together the 1 cup all-purpose flour 1 1/4 cup dry quick oats, 1/4 teaspoon cinnamon and 1/4 teaspoon nutmeg1/2 teaspoon baking soda, and 1/4 teaspoon kosher salt in a separate bowl. Add the dry ingredients to the stand mixer and mix until well combined.
Line the air fryer basket with cookie parchment paper, add 6-8 cookie dough balls on top, and use your hands to flatten the top of the cookie dough.
Set the air fryer temperature to 320 degrees F, and set the cook time for 4-6 minutes. Then, allow the cookies to cool in the air fryer basket for at least 5 minutes before transferring them to a cooling rack.
While the cookies cool, make the icing. Using a spoon or a whisk, beat the 3/4 cup powdered sugar 2 tablespoons milk and 1/4 teaspoon vanilla extract together in a small bowl.
Dip the tops of the cookies into the icing, allowing the excess icing to drip back into the bowl. Then sprinkle the cookies with a small pinch of cinnamon and quick oats. Let the cookies harden and set for at least 5 minutes before serving.