Center an oven rack and heat the oven to 325 f. Grease and flour two 9-inch cake pans. Line the bottoms with parchment paper circles. Set aside.

In a medium bowl, whisk the flour, baking powder, baking soda, and salt together.

Melt the butter in a large saucepan over medium heat. Stir in the cocoa and cook until fragrant, about a minute. Off the heat, whisk in the coffee, buttermilk, and sugars until dissolved. Whisk in the eggs and vanilla, then slowly whisk in the flour mixture. Divide the batter evenly between the prepared cake pans and smooth with a spatula.

Bake until a toothpick inserted into the centers comes out clean, about 30-35 minutes, rotating the cake pans halfway through baking. Cool the cakes in the pans for 15 minutes, then invert onto a wire rack. Cool the cakes completely before frosting.

Make the filling:

Cook the sugar, chocolate, half-and-half, milk, cornstarch, and salt in a large saucepan over medium heat, whisking constantly, until the chocolate is melted and the mixture is smooth, shiny and thickened. It took me about 20-22 minutes to get the right consistency of the pudding comparing to 2-4 minutes suggested time. So, consider the time as a guidance and follow your own judgment. Off the heat, stir in the vanilla. Transfer the pudding to a large bowl and refrigerate, with plastic wrap pressed flush against its surface, until cold and set, at least 4 hours or up to 24 hours.

Assemble the cake:

Using a large serrated knife, slice each cake into 2 even layers. Crumble one cake layer into medium crumbs and set aside. Place one cake layer on a cake plate or cardboard round. Spread 1 cup of the pudding over the cake layer and top with another layer. Repeat with 1 cup more pudding and the last cake layer. Spread the remaining pudding evenly over the top and sides of the cake. The whole construction might look a bit jiggly at this point but after you press the crumbs all around it will get more stable, especially after some refrigeration time. Sprinkle the cake crumbs evenly over the top and sides of the cake, pressing lightly to adhere the crumbs.

The cake can be made a day ahead and kept under a cake dome in the refrigerator.