Use an electric whisk to beat the eggs and sugar in a large bowl until creamy. Gradually add water and oil while continuing to beat until the mixture is even and slightly fluid. Mix in the flour, cocoa powder, and baking powder. Divide the batter into two equal portions on deep baking trays lined with parchment paper and smooth the surface.
Mix the cream cheese, egg, icing sugar, and vanilla pudding powder with an electric whisk. Fill a piping bag with the mixture and create a grid-like pattern on one of the trays of batter.
Bake both portions of batter in a preheated convection oven at 360°F for 30 minutes. Let them cool afterwards.
Heat 17 fl oz of milk with sugar in a large pot. Stir the chocolate pudding powder and cornstarch into the remaining milk, then add this mixture to the hot milk while stirring continuously to make pudding. Remove from heat and let it cool.
Beat the butter with icing sugar using an electric whisk and mix in the cold chocolate pudding to make buttercream. Spread half of the pudding filling onto the cake without the pattern.
Cut the bananas in half and place them on the pudding layer. Spread another layer of pudding on top of the bananas, then place the patterned cake piece on top.
