Pulse the oats in a food processor 10-12 times until you have a variety of texture– chopped oats with some oat flour.
Whisk the pulsed oats, flour, baking soda, salt, ginger, cinnamon, nutmeg, and cloves together in a medium bowl. Set aside.
In a large bowl using a hand mixer or a stand mixer fitted with a paddle attachment, beat the butter, brown sugar, and granulated sugar together on medium-high speed until creamed, about 3 minutes. Add the egg and molasses and beat on high speed until combined, about 1 minute.
Add the dry ingredients to the wet ingredients and mix on low until combined. Chill the dough for 30-45 minutes in the refrigerator (and up to 4 days).
Preheat oven to 350°F (177°C). Line large baking sheets with parchment paper.
Scoop cookie dough, about 1.5 Tablespoons per cookie, and place 3 inches apart on the baking sheets. Bake for 12-13 minutes or until lightly browned.
Remove from the oven and allow cookies to cool on the baking sheet for 5 minutes before transferring to a wire rack.
Combine confectioners’ sugar, vanilla extract, and 1 Tablespoon of milk in a medium bowl. Whisk until combined, adding more milk for desired thickness.
Lightly dip the tops of the cookies into the icing or drizzle icing on top. Dust/sprinkle more ground cinnamon on top if desired.
Cookies stay fresh covered at room temperature for 3 days or in the refrigerator for up to 10 days.