Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a mixing bowl, cream together the softened butter, brown sugar, and granulated sugar. Add the egg and vanilla extract, then mix until light and fluffy.
Stir in the rolled oats, all-purpose flour, cinnamon, nutmeg, baking soda, and salt until well combined.
Scoop out the cookie dough onto the prepared baking sheet, flatten slightly, and bake for 10 minutes.
Allow the cookies to cool on the baking sheet for 20 minutes before transferring them to a cooling rack to cool completely.
For the vanilla sugar glaze, whisk together the powdered sugar, milk, and vanilla until smooth. Adjust the consistency by adding more milk if needed.
Dip the cooled cookies into the glaze and place them back on the cooling rack to set before serving.
Notes
For added texture, you can sprinkle some extra oats on top of the glaze before it sets.