In a medium bowl, whisk together the oats, flour, cinnamon, nutmeg, baking soda, and salt. The oats should be evenly distributed with no dry flour pockets hiding at the bottom.

In a large bowl, beat the softened butter, brown sugar, and granulated sugar on medium speed until light and fluffy, about 3 minutes. Scrape down the bowl twice - you want the mixture to look like wet sand that holds its shape when you press it.

Beat in the egg and vanilla until fully incorporated, about 1 minute. The mixture will look slightly curdled at first, then smooth out.

Add the oat mixture to the butter mixture and stir on low speed just until no dry streaks remain, about 30 seconds. The dough will be thick and slightly sticky. Cover and chill for 15 minutes while you preheat the oven.

Preheat oven to 350F. Line two baking sheets with parchment. Scoop 2-tablespoon mounds of dough, roll into rough balls, and place 2 inches apart. Bake until the edges are set and the centers still look slightly underdone, 11 to 13 minutes. The tops will crack and look dry - that's exactly what you want.

Let cookies cool on the baking sheets for 5 minutes, then transfer to a wire rack to cool completely. They will firm up as they cool. Do not ice warm cookies or the glaze will melt off.

Whisk together the powdered sugar, milk, and vanilla until smooth and pourable but thick enough to hold a line. It should ribbon off the whisk and disappear back into the bowl in 2 to 3 seconds. Add milk 1/2 teaspoon at a time if too thick.

Dip the tops of completely cooled cookies into the icing, letting excess drip off, or spoon and spread about 1 teaspoon on each. Set back on the rack and let the glaze set until crackled and firm to the touch, about 30 minutes.