Preheat your oven to 350°. Butter 2 (9-inch) round cake pans. Line with parchment paper, then butter and flour the pans, tap out any excess flour.

In the bowl of your mixer; sift the flour, sugar, cocoa, baking soda, baking powder, and salt. With the paddle attachment in place, mix on low speed until combined. In another bowl, combine the buttermilk, applesauce, eggs, and vanilla. With the mixer on low speed, slowly add the wet ingredients to the dry. With mixer still on low, add the cooled coffee and stir just to combine, scraping the sides and bottom of the bowl with a spatula to get all the ingredients incorporated. Pour the batter into the prepared pans and bake for 35 to 40 minutes, rotating them half way through, until a cake tester comes out clean. Cool in the pans for 25 minutes, then turn them out onto a cooling rack and cool completely.

Place 1 layer, upside down, on a flat plate or cake pedestal. With a knife or offset spatula, spread the top with about a third of the frosting. Place the second layer on top, right-side up, and spread the frosting evenly on the top and sides of the cake.

Roughly chop the chocolate and place it in a heat-proof bowl set over a pan of simmering water. Stir until just melted and then set aside to cool down to room temperature.

In the bowl of your mixer fitted with the paddle attachment, beat the butter on medium-high speed until light and fluffy, about 2-3 minutes. Add the beaten egg yolk and vanilla and continue beating for 2-3 minutes. Turn the mixer to low, working in batches, gradually add the confectioners' sugar, and then beat at medium speed, scraping down the sides and bottom of the bowl as necessary, until smooth and creamy. Reduce the speed to low, add the cooled chocolate to the butter mixture and mix until blended. Careful not to whip! Spread immediately on the cooled cake.