Grease a baking pan with 1 Tablespoon solid fat, then dust it with 2 Tablespoons fine breadcrumbs or all-purpose flour. Turn the pan onto a work surface and gently tap to remove any excess flour.
In a separate bowl, combine dry ingredients: 1 ⅔ cup all-purpose flour, 1 Tablespoon cocoa powder, and 1 ½ teaspoons baking powder.
In a mixing bowl, use an electric handheld mixer to whisk 3 eggs on medium speed, gradually adding 1 ½ cups powdered sugar until a foamy mixture forms. Once all the sugar has been added, increase the speed and continue whisking. Be patient—after about 5 minutes, you will achieve a light and fluffy foam.
Reduce the mixer speed to low and gradually add the prepared flour mixture, ½ cup milk, and ½ cup vegetable oil to the beaten eggs. Mix until everything is combined into a semi-liquid, light brown sponge cake batter.
Pour the batter onto the prepared baking pan and smooth the surface.
Cook chocolate pudding according to the manufacturer's instructions. I prepare pudding by gently heating 3 cups milk, whisking in 2 packages Dr. Oetker chocolate pudding powder and 4 Tablespoons granulated sugar, and bringing it to a boil over medium heat, stirring constantly to prevent lumps from forming.
Pour the hot pudding in a zigzag pattern over the batter, leaving some gaps so the underlying batter is visible. Avoid covering the surface uniformly.
Preheat the oven to
350 °F (lower and upper heating). Put the cake in the oven and bake for 30 minutes. The cake will puff up as it bakes. While the cake is baking, prepare the final layer by whisking 1 ⅔ cups sour cream with 3 Teaspoons powdered sugar. Once the cake is out of the oven, immediately pour the sour cream mixture on top and smooth the surface with a spoon or an offset spatula. The heat will thin the sour cream, allowing it to spread evenly over the entire surface.
Let the cake cool at room temperature. Then, place it in the fridge, best overnight. This will allow the sour cream to firm up, ensuring it won't run when slicing the cake.