Brown the butter. Place the butter in a medium saucepan and melt over medium heat, stirring often, until it foams, smells nutty, and browns. This will take about 5 minutes. Scrape the browned butter into a large mixing bowl and set aside to cool to room temperature. You can put the bowl in the fridge to speed up the cooling process.
Make the cookie dough. Add the oats to a food processor and pulse about 8 times. You want a mix of chopped oats and oat flour.
In a medium bowl, combine the oats, flour, baking soda, salt, cinnamon, and nutmeg. Stir and set aside.
Add the sugars, egg, egg yolk, and vanilla to the browned butter. Use a spatula to stir until ingredients are well combined.
Add dry ingredients and stir until just combined. Don’t over mix. Chill the dough for 30 minutes. You can chill for up to 72 hours.
When ready to bake, preheat the oven to 350 degrees F. Remove the dough from the fridge and let come to room temperature while the oven preheats.
Scoop the cookie dough into balls, about 1 1/2 tablespoons of dough per cookie, and place about 3 inches apart on baking sheets that have been lined with parchment paper.
Bake for 10 to 12 minutes or until lightly browned on the edges, but still soft in the centers. Don’t over bake. Remove the cookies from the oven and let cool on the baking sheet for 5 minutes. Transfer to a cooling rack to cool completely.
Make the icing. In a medium bowl, whisk together the confectioner’s sugar, 3 tablespoons eggnog, vanilla, and salt. Whisk until the icing is smooth. You want the icing to be thick, but if it is too thick, add 1 more tablespoon of eggnog and whisk again.
Dip the tops of the cooled cookies into the icing and let the excess drip off. Set the dipped cookies on a wire cooling rack and let sit until the icing sets up.