In a bowl, combine the cooled coffee and rum or brandy (if using). Dip half of the gingerbread cookies into the coffee mixture, making sure they are soaked but not overly saturated. Place the soaked cookies in a single layer at the bottom of a 9×13-inch baking dish.
In a mixing bowl, whisk together the eggnog, heavy cream, sugar, vanilla extract, cinnamon, and nutmeg until well combined and slightly thickened.
Pour half of the eggnog mixture over the layer of soaked gingerbread cookies in the baking dish. Use a spatula to spread it evenly.
Dip the remaining gingerbread cookies into the coffee mixture and arrange them in a second layer over the eggnog mixture.
Pour the remaining eggnog mixture over the second layer of gingerbread cookies, spreading it out evenly.
Cover the dish with plastic wrap and refrigerate for at least 4 hours or overnight to allow the flavours to blend and the cookies to soften.
Before serving, dust the top of the tiramisu with cocoa powder. Cut into squares or scoop out portions and serve chilled.