
Preheat the oven to 350°F.
In a large microwave-safe bowl, microwave the butter in 20-second bursts until melted. Set aside to cool to room temperature (about 70°F).
In a food processor, pulse the oats 15 times, or until broken up into the consistency of coarse crumbs. Add flour, cinnamon, salt, baking powder, baking soda, and nutmeg. Pulse 2-3 times to combine.
To the large bowl containing the melted and cooled butter, whisk in the brown sugar. Add the eggs, egg yolk, and vanilla, and whisk until very well combined. Stir in the dry ingredients with a spatula until just combined.
Let sit for 5 minutes to hydrate (10-15 minutes in humid climates).
Line baking sheets with parchment paper. Using a medium, spring-loaded scoop, drop 1 1/2-tablespoon balls of dough onto baking sheets.
Bake for 10-11 minutes, or until the edges are lightly browned and the centers are set. Don’t underbake, or they will be too delicate to ice.
Right after removing from the oven, use a round cookie or biscuit cutter slightly larger than the cookies to swirl in circles around each cookie to gently reshape them into thick circles.
Let cookies cool on baking sheets for 5 minutes before removing to wire racks to cool completely.
In a medium shallow bowl or dish, combine all the icing ingredients with a small whisk or fork until a very thick icing forms. It should fall off the whisk in large ribbons. Add extra milk only a splash at a time if needed, so the icing doesn’t get too thin.
Lightly dab each cooled cookie into the icing, avoiding dunking completely so the cookie still peeks through. Place back onto the parchment-lined baking sheets.
Let icing set for about 1 hour before stacking, transporting, or storing. Cookies can be stored in an airtight container at room temperature for up to 3 days.