Using a stand mixer fitted with the paddle attachment, cream together the room temperature brown butter and powdered sugar until fully combined.
Then add in the egg yolk, eggnog, vanilla, and rum extract and mix until fully combined and fluffy.
In a separate bowl, sift together the flour, almond flour, baking powder, nutmeg, cinnamon, and salt.
Gently fold in the dry ingredients until the dough comes together into a ball. It should be very soft, but not sticky.
Divide in half and press the dough into discs. Wrap tightly in plastic wrap and refrigerate overnight (or a minimum of at least 2 hours).
Line 2 baking sheets with parchment paper and set aside.
On a lightly floured surface, roll out the dough to about ¼ inch thick then cut out the cookies into your desired shapes. (For crispier cookies you can roll the dough thinner. For chewier cookies roll to about ½ inch instead.)
Place the eggnog cookies 2 inches apart on your lined baking sheet. Place the baking sheet in the fridge or freezer for a minimum of 20 minutes before baking. This ensures the cookies keep their shape.
While the cookies are chilling, preheat your oven to 350°F.
Bake for 8-12 minutes or until the edges are set. Bake on the middle rack of the oven and rotate the pan halfway through baking for the best results.
Let the cookies cool on the baking sheet for 2-3 minutes, then remove and let them finish cooling on a cooling rack. Let the cookies cool completely before frosting.
Eggnog Frosting
In a large bowl using a handheld or stand mixer fitted with a paddle attachment, cream the brown butter until smooth.
Add in the powdered sugar ½ cup at a time until fully combined and smooth.
Slowly drizzle in the eggnog until fully incorporated. Then add in the rum extract if desired. Beat on high for 3-4 minutes or until the buttercream is light and fluffy.
Dollop about 1 tablespoon of eggnog frosting on top of the cookies and spread evenly with an offset spatula. Then garnish with sprinkles if desired!
Notes
Store in an airtight container in the fridge for up to a week.To make brown butter, melt unsalted butter over medium heat, stirring frequently. Watch closely for the butter to develop brown specs and a nutty aroma. Once you start seeing brown specs, let cook for just a few seconds longer (watch very carefully or it will burn). Remove from the heat and pour into a small bowl so that it doesn't continue to cook. Set aside and let cool for about 20 minutes in the fridge until soft and pliable. Note: For every ½ cup (1 stick or 113g), you'll lose about 1 tablespoon (14g) to evaporation while cooking. So you can either add in an extra 1 tablespoon before browning, or if you're browning big batches, just make up the weight difference in your recipe with regular unsalted butter if needed.