For the Pudding: Set a fine-mesh strainer over a large bowl and set aside. In a large saucepan, whisk together sugar, cornstarch, cocoa powder, and salt until well combined. Gradually whisk in milk and cream until smooth. Set saucepan over medium heat and cook, whisking constantly and scraping the bottom and sides of the pan, until mixture thickens and begins to gently bubble, about 5 to 7 minutes. Once bubbling, continue to cook while whisking until cornstarch is fully cooked and no raw, starchy flavor remains, about 30 seconds. Remove saucepan from heat. Immediately add unsweetened chocolate and butter and whisk until chocolate is completely melted and smooth.

Immediately strain pudding through prepared fine-mesh strainer, using a flexible rubber spatula to press pudding through; discard any solids. Whisk in vanilla extract until incorporated. Press plastic wrap or parchment paper directly onto surface of pudding to prevent a skin from forming. Refrigerate until fully chilled and set, at least 4 hours or up to 3 days.

For the Cake: Adjust oven rack to middle position and preheat oven to 350°F (175°C). Coat two 8-inch round cake pans with cooking spray and line bottoms with parchment paper rounds; set aside.

In a medium bowl, whisk together cocoa powder, flour, espresso powder, salt, baking soda, and baking powder until well combined; set aside.

In a large bowl, whisk together brown sugar, granulated sugar, sour cream, hot water, vegetable oil, eggs, and vanilla extract until smooth. Add dry ingredients and whisk just until combined and no dry pockets remain (batter will be loose).

Divide batter evenly between prepared pans and smooth tops with an off-set spatula. Bake until cakes are set and a toothpick or cake tester inserted into the center comes out with a few moist crumbs, 25 to 32 minutes.

Transfer pans to a wire rack and let cakes cool completely in pans, 1 to 1½ hours. Once cooled, invert cakes onto rack, remove parchment, and invert again so cakes are right side up. Cakes can be wrapped tightly and stored at room temperature for up to 2 days before assembling.

To Assemble Cake: Stir pudding briefly to recombine. Using a serrated knife and placing one hand flat on top of the cake to steady it, split each of the cakes horizontally, creating 4 layers total. Crumble one cake layer into fine crumbs and set aside.

Place one cake layer on a serving platter or cardboard cake round. Using an off-set spatula, spread about 1 cup (240 ml) pudding evenly over cake layer, then top with a second cake layer. Spread about 1 cup (240 ml) pudding over second layer and top with a third cake layer. Spread remaining pudding evenly over top and sides of cake. Sprinkle reserved cake crumbs evenly over top and sides, pressing gently to help crumbs adhere. For cleaner slices, refrigerate cake, uncovered, for 1 hour to allow pudding to firm up before serving. After cake is chilled, slice and serve immediately. Alternatively, cover unsliced cake loosely with a cake dome and refrigerate until ready to serve.