For the cake:
Heat the oven to 350°F and arrange a rack in the middle. Coat a 13-by-9-inch baking dish with butter; set aside.
Fill a small saucepan with 1 inch of water and bring it to a simmer over medium-high heat. Place the chocolate in a stainless steel or glass bowl, set the bowl over the simmering water (be sure it’s not touching the water), and stir until the chocolate is completely melted, about 3 to 5 minutes; set aside to cool slightly.
Combine the flour, cocoa powder, baking soda, and salt in a large bowl and whisk to aerate and break up any lumps; set aside.
In the bowl of a stand mixer fitted with a paddle attachment, cream the measured butter and sugar on medium-high speed until light in color and fluffy, about 4 minutes. Add the eggs, 1 at a time, then the egg yolk, beating after each addition until evenly incorporated. Stop the mixer and scrape down the sides of the bowl using a rubber spatula. Add the vanilla and melted chocolate and return the mixer to medium-high speed, mixing until evenly combined, about 30 seconds.
Reduce the speed to medium low and add a third of the reserved dry ingredients. When just incorporated, add half of the buttermilk. Repeat, scraping down the bowl at least once and ending with the last third of the dry ingredients.
Pour the batter into the prepared dish, smooth the top with the spatula, and bake until a toothpick inserted in the middle comes out with only a few crumbs, about 35 minutes. Cool completely in the pan on a wire rack, about 1 hour. Meanwhile make the filling.
For the filling:
Combine the cream, sugar, and egg yolks in a medium saucepan over medium heat, stirring constantly until the mixture thickens and coats the back of the spoon, about 8 minutes.
Transfer to a medium bowl, add the remaining ingredients, and stir until the butter has melted and is evenly incorporated. Let cool to room temperature.
Once the cake has cooled, carefully transfer it to a cutting board and slice it into thirds (the pieces should measure about 4 inches wide by 9 inches long), trimming as necessary to make the sections as even as possible.
Spread about a third of the filling evenly over the top of 1 piece of cake, and another third of the filling over the top of a second piece. Stack the 2 pieces and place the last, unadorned piece on top. Set the last third of the filling aside.
Using about a third of the vanilla buttercream, frost the sides and top of the cake. (Don’t worry about looks at this point—this is just a base coat, or crumb layer, and it will be covered up later.) Place the cake in the freezer for about 10 minutes, or the refrigerator for about 20 minutes, until the frosting is set up and slightly hard. Evenly spread the remaining frosting over the top and sides of the cake, then evenly spread the remaining third of the filling on top of the cake.
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