First, preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper or silicone mats. To ensure the best texture, make sure your butter, egg, sour cream, and cream cheese are all softened to room temperature.
In a medium bowl, gently whisk together the flour, cocoa powder, baking soda, and salt until well combined. Set this mixture aside for a moment.
In a large mixing bowl, using an electric mixer with a paddle attachment, beat the softened butter and granulated sugar on medium-high speed. Continue for about 2 minutes until the mixture looks light and fluffy.
Now, beat in the room temperature egg until fully incorporated. Then, add the 1 tablespoon of eggnog (or milk), your chosen red food coloring, vanilla extract, and the optional peppermint extract. Mix briefly until everything is just combined, about 30 seconds.
Reduce your mixer speed to low. Gradually add about half of your dry flour mixture, followed by the sour cream, and then the remaining half of the flour mixture. Mix just until combined – be careful not to overmix the dough to keep your cookies tender!
Using greased hands or a cookie scoop (about 2 tablespoons per cookie), roll the dough into smooth balls. Place them on your prepared baking sheets, leaving approximately 2 ½ inches between each cookie. Bake for 9-11 minutes, or until the edges are lightly set and the centers look barely cooked. They might seem soft, but they'll firm up beautifully! Let the cookies cool on the baking sheet for 5 minutes before carefully transferring them to a wire rack to cool completely.
While the cookies are cooling, prepare the dreamy frosting! In a clean mixing bowl, beat the softened cream cheese, softened butter, 2 tablespoons of eggnog (or milk), and vanilla extract until the mixture is wonderfully smooth and creamy.
Add the cinnamon and nutmeg to the frosting, then gradually beat in the powdered sugar until the frosting is light and fluffy. If your frosting seems too thick, you can add a tiny bit more eggnog; if it's too thin, a little more powdered sugar will do the trick!
Once your cookies are completely cool, spread a generous dollop of cream cheese frosting on top of each. Feel free to add sprinkles or crushed candy canes for a fun, festive touch!