Heat oven to 350ºF. Line the bottoms of two 9-inch cake pans with parchment paper. Grease the paper, then lightly coat with cocoa powder.

Add the flout, sugar, cocoa powder, baking soda, baking powder, and salt to the bowl of an electric mixer fitted with a whisk attachment. Mix on low speed for 10-15 seconds, just long enough to combine the ingredients and break up and clumps. Pour in the yogurt, oil, eggs, vanilla, and coffee. Stir on low speed just long enough to combine the ingredients into a thin batter.

Pour the batter into the prepared cake pans. Bake for 30-35 minutes, or until a cake tester inserted into the center comes out clean (the center of the cake might not appear done, but check it anyway.) Remove the pans from the oven and let the cakes cool for 30 minutes, or until the edges begin to pull away from the pan. Run the edge of a knife around the pan to loosen, then invert onto a cooling rack. Cool cakes completely (If time allows, wrap each layer in plastic and refrigerate overnight).

In a small saucepan set over medium heat, whisk together the flour and eggnog. Cook until very thick and paste-like, about 5 minutes. Let cool completely. Stir in the rum flavoring.

Add the butter and sugar to the bowl of a stand mixer fitted with a paddle attachment. Beat until light and fluffy. Add the cooled eggnog mixture; beat on high speed until very light and fluffy – 3-5 minutes.

If necessary, trim the tops of the cake layers to flatten them. Place one layer on a cake stand or serving plate. Spread the top of the cake with 1 1/2 cups of icing, making the layer as even as possible. Place the second cake layer on top. Use about 2 cups of frosting to cover the cake in a very thin layer (crumb coat). Let sit at least 15 minutes to allow the frosting to harden. Use the rest of the frosting to frost the cake as desired.