Adjust oven rack to lower-middle position and preheat oven to 350 degrees F.
Prepare the Pan: Stir together the melted butter and 1 tablespoon flour in a small bowl until a paste forms. Using a pastry brush, coat all of the interior surfaces of a 12-cup Bundt pan.
Make the Cake: In a medium bowl, whisk together the flour, salt, baking powder and baking soda; set aside. In a small bowl, whisk together the eggnog and vanilla; set aside.
Using an electric mixer, beat the butter and sugar on medium-high speed until pale and fluffy, about 3 minutes. Add the eggs and the yolk, one at a time, and beat until combined, scraping the sides of the bowl as needed. Reduce the mixer speed to low and add the flour mixture in 3 additions, alternating with the eggnog mixture in 2 additions, scraping the bowl as needed.
Combine 1 cup of the cake batter with the 1 tablespoon of rum, nutmeg and cinnamon in a medium bowl, and stir until just combined. Spread the remaining batter in an even layer in the prepared pan. Spoon the spiced batter over top and swirl gently with a butter knife.
Bake until a toothpick inserted in the center comes out with a few moist crumbs attached, 50 minutes to 1 hour. Let the cake cool in the pan for 25 minutes, then turn it out onto a wire rack to cool completely, about 3 hours.
Make the Rum Icing: In a medium bowl, whisk together the powdered sugar and remaining 3 tablespoons of rum until smooth. The icing should be a thick pouring consistency; if it is too thick to pour, add eggnog 1 tablespoon at a time, until the appropriate consistency is reached. Pour the icing over the cooled cake and allow to set for at least 30 minutes before serving. The cake can be stored at room temperature, wrapped tightly in plastic wrap, for up to 4 days.