Pre-heat oven to 350°F (177°C). Grease a 13 inch (33cm) by 9 inch (23 cm) cake pan with the softened butter and dust the inside with the sugar.
In a bowl combine the oil, molasses and eggs, then set aside. In a large mixing bowl, combine the flour, baking soda, ginger, cinnamon, cardamom and salt.
In a small pot bring the brown sugar and root beer to a simmer to dissolve the sugar. Add the molasses mixture to the flour mixture and mix to combine. Carefully add the hot root beer mixture and quickly beat until combined, scraping down the bowl during mixing.
Pour the batter into the prepared cake pan and bake in the centre of the oven until a toothpick comes out clean, approx. 35-40 minutes. Remove from the oven, let cool in the pan on a rack at least 15 minutes. Invert the pan on a plate until cake releases from pan. Slice and serve topped with the Eggnog Custard Sauce and a sprinkle of cinnamon.
In a heavy sauce pan combine the eggnog and the vanilla, and bring to a gentle simmer over medium-low heat.
In a bowl whisk the egg yolks and sugar until smooth. Whisk in a touch of the hot eggnog, then add the entire egg mixture into the remaining hot eggnog.
Stir constantly over medium-low heat until the sauce coats the back of a spoon, approx. 8 minutes. Place a layer of plastic directly on the surface of the custard and refrigerate until needed, up to two days. Serve warm with the Gingerbread Cake.