Preheat oven to 325°F. Grease a springform pan and add parchment paper, if using.
Warm the eggnog in the microwave or on a stovetop until lukewarm (between 1-2 minutes). Whisk in the rum and vanilla extract. Set aside.
In a separate bowl, whisk the egg whites and cream of tartar in a stand mixer or with a hand mixer for 4-5 minutes, until stiff peaks form. Set aside.
In a stand mixer or using a hand mixer, beat the egg yolks with the confectioners sugar on medium high speed until thick and pale yellow, which usually takes 2-3 minutes.
Add in the cooled melted butter and beat until fully incorporated.
Add the flour into the batter in three parts, making sure to scrape down the sides of the bowl each time. On low speed, slowly pour in the eggnog mixture until combined.
By hand, fold the egg whites into the cake batter, making sure not to knock any air out of the batter. The batter will look loose with a few lumps of egg white.
Pour the batter into the baking pan and bake until the top is golden brown and there is a slight jiggle to the center, about 55 minutes to 1 hour.
Allow the cake to cool completely on a wire rack and release from the springform pan or remove from the round, square, or bundt cake pan.