Preheat the oven to 350 degrees and line baking sheets with parchment paper.
Add the old-fashioned oats to a food processor or blender and pulse until you have a mix of finely ground oats and small oat pieces. About 10 to 12 pulses should do it, but go by texture rather than the exact number. Do not process into a fine oat flour.
In a medium bowl, whisk together the pulsed oats, flour, cornstarch, baking powder, baking soda, salt, cinnamon, and nutmeg. Set aside.
In a large mixing bowl or the bowl of a stand mixer, cream the butter, brown sugar, sugar, and vanilla until light and fluffy, about 2 minutes. Mix in the egg on low speed until combined. Gradually add the dry ingredients and mix just until combined. Do not overmix.
Cover and refrigerate the dough for 20 to 30 minutes. The dough will be soft and slightly sticky, and chilling gives the oats time to absorb moisture while helping the cookies spread less in the oven.
Scoop the dough into 1½-tablespoon portions and place about 2 inches apart on the prepared baking sheets. Bake closer to 10 to 11 minutes for softer cookies and 14 to 15 minutes for a crisper cookie. The edges should look set and lightly golden while the centers still look slightly soft. The cookies will continue to set as they cool.
Let the cookies cool on the baking sheet for about 5 minutes before transferring them to a wire rack to cool completely.
Whisk the powdered sugar, vanilla, and milk until smooth. The glaze should be thick enough to coat the tops of the cookies without running into every crevice. If it is too thin, add a little more powdered sugar. If it is too thick, add milk 1 teaspoon at a time.
Dip just the tops of the completely cooled cookies into the glaze, then lift straight up and let the excess drip off. Leave some of the craggy oatmeal texture showing through for that classic iced oatmeal cookie look.
Let the glazed cookies sit uncovered until the icing is firm. Give them at least 30 minutes before serving and longer before stacking or packaging.