Preheat your oven to 350°F. Line baking sheets with parchment paper.
Place the rolled oats in a food processor and pulse 10-12 times until coarsely ground.
In a medium bowl, whisk together the ground oats, flour, baking soda, baking powder, cinnamon, nutmeg, and salt.
In a large bowl, beat the softened butter, brown sugar, and granulated sugar until light and fluffy.
Beat in the eggs, vanilla extract, and molasses until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Scoop the dough into balls about 2 tablespoons each and place them on the prepared baking sheets, spacing them 2 inches apart.
Bake the cookies for 10-12 minutes or until the edges are lightly golden. Allow the cookies to cool on the baking sheets for 5 minutes, then transfer them to wire racks to cool completely.
In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth. Start with 1 tablespoon of milk and slowly add more until the icing reaches a consistency you can dip the cookies in.
Dip the tops of the cooled cookies into the icing and let the excess drip off.
Place the cookies back on the wire racks and allow the icing to set before serving.