Pulse oats in a food processor until varied in texture.
Whisk together pulsed oats, flour, baking soda, salt, cinnamon, and nutmeg.
Cream butter and sugars, then add eggs, molasses, and vanilla extract.
Combine wet and dry ingredients, then chill the dough.
Preheat oven to 350°F and prepare baking sheets.
Scoop dough onto baking sheets and bake until edges are lightly browned.
Cool cookies on sheets, then transfer to a wire rack.
Prepare icing and dip the tops of the cookies into it.
Store cookies at room temperature or refrigerated.