Preheat the oven to 350F. Line 2 large baking trays with parchment paper or silpat liners.
Add the flour, oats, cinnamon, nutmeg, baking soda, and salt to a medium bowl. Stir to combine.
Use a handheld electric mixer to beat together the butter, brown sugar, and granulated sugar in a large bowl. Beat in the egg and vanilla.
Add the dry ingredients to the butter mixture and beat to combine.
Stir in the diced apple.
Use a 1 1/2-tablespoon scoop to measure out the dough. Roll it into balls and arrange them about 2 inches apart on the prepared baking trays. Slightly flatten each cookie.
Bake until the cookies are golden on the bottom and set along the outside but still look a touch doughy in the center. This takes about 13 to 14 minutes if baking 2 trays at once (rotate them once halfway through), or 11 to 12 minutes if baking 1 tray at a time.
Let the cookies cool for 10 minutes on the trays, and then use a thin metal spatula to transfer them to a wire rack to finish cooling.
Once the cookies are cooled, make the icing. To do so, use a fork to stir enough milk into the powdered sugar so that it reaches a glaze-like consistency.
Drizzle the icing decoratively on top of the cooled cookies.
Let the icing set before serving the cookies.
Notes
Can I freeze this dough? Oatmeal cookie dough can be frozen and baked later. To do that, line a cookie sheet with wax paper and scoop the dough into similar-sized balls of dough for even cooking. After an hour, remove the frozen dough from the trays and add them to a labeled freezer bag. Force out all the air, then store them in the freezer for up to 6 months.