In a saucepan, add all but ¼ cup of the milk. Take the remaining milk and mix it well with the cornstarch. Add the mixture to the milk. Heat on medium, stirring frequently and scraping the bottom and sides of the pan.
Add the salt, sugar, egg, and yolk to a bowl and beat well until it is pale and ribbony.
Once the milk mixture is hot and nearly boiling, drizzle one cup of the milk into the egg mixture while stirring the egg mixture.
Repeat with another cup of the hot milk.
Transfer the egg mixture and the heavy cream into the saucepan and stir it in well.
Continue to cook, stirring frequently until the mixture is visibly thickened.
Remove from heat and stir in the butter and extract.
Pour into a container and place plastic wrap directly touching the pudding. Chill until cooled - at least 5 hours.
Make the Cake:
Preheat the oven to 350°F. Grease and flour a 9 x 13 baking pan. In a medium bowl, mix together the flour, baking powder, baking soda, and salt. Mix and set aside.
In a stand mixer, beat the butter and sugar until light and fluffy. About 4 minutes.
Add in the eggs and beat well.
Stir in one third of the flour mixture, then half the buttermilk and 1 1/2 tablespoons of vanilla. Add in half the remaining flour and the remaining milk. Stir until just combined. Mix in the remaining flour.
Pour into the baking pan and spread evenly. Bake for about 30 - 40 minutes, or until a toothpick comes out clean when inserted.
Remove and allow to cool completely.
Assemble the Cake:
Poke holes on the top of the cake using the bottom of a wooden spoon. Pour over half the pudding over the cake, ensuring it gets into the holes. Smooth the pudding over the cake, adding more as needed to get a nice even layer.
Top with sliced bananas.
Whip the heavy cream with the powdered sugar, salt, and extract. Spread the whipped cream over the bananas.