Preheat the oven to 350°F. Line a baking sheet with parchment paper.
In a mixing bowl, beat the butter, brown sugar, and granulated sugar until creamy. Add the egg and vanilla and mix until combined.
½ cup butter - softened and at room temperature, ⅓ cup brown sugar - substitute with coconut sugar if desired, ¼ cup sugar, 1 egg - room temperature, 1 teaspoon vanilla extract
Stir in the oat flour, rolled oats, cinnamon, nutmeg, baking soda, and sea salt until just combined.
1 cup oat flour - I used Bob’s Red Mill, ½ cup quick rolled oats, 1 teaspoon cinnamon, ¼ teaspoon nutmeg, ½ teaspoon baking soda, ½ teaspoon sea salt
Scoop the dough onto the prepared baking sheet, spacing the cookies about 2 inches apart.
Bake for 9–11 minutes, until the edges are set and lightly crisp and the centers are just set. Cool completely.
In a shallow bowl, whisk together the powdered sugar and milk to make the icing (it will be thick). Dip the tops of the cooled cookies into the icing, letting excess drip off.
1 cup powdered sugar, 1 tablespoon milk
Place cookies on a rack or baking sheet and let the icing set before serving.