Preheat your oven to 350˚F. Grease a 9-inch round cake pan with butter and line it with parchment paper.
In a medium bowl, whisk together the flour, baking powder, and salt until combined.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and sugar on medium speed until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition. Scrape down the sides of the bowl as needed.
Add half of the flour mixture to the butter mixture and mix on low speed until just combined. Add in the buttermilk and mashed banana and mix until just combined. Add in the remaining flour mixture and mix until just combined.
Pour the batter into your prepared cake pan and spread evenly. Bake for 35-40 minutes, or until a toothpick inserted in the center comes out clean. Let cool completely before topping with custard.
To make the custard, whisk together the sugar, cornstarch, and salt in a medium saucepan.
Add the milk to the sugar mixture and stir until combined. Place the pan over medium heat and bring to a simmer, stirring constantly. Simmer for 1 minute, then remove from heat.
In a small bowl, whisk together the egg yolks. Slowly add a few tablespoons of the hot milk mixture to the yolks and whisk until combined. Slowly add the egg yolk mixture back into the saucepan, stirring constantly. Return the pan to medium heat and bring to a simmer, stirring constantly until thickened about 3 minutes.
Remove from heat and stir in the butter and vanilla extract. Pour custard into a bowl and place plastic wrap over top, pressing it onto the surface. Refrigerate for at least 3 hours, or until cooled completely.
To make the whipped cream, beat together the heavy cream, confectioners’ sugar, and vanilla extract until stiff peaks form.
To assemble the cake, spread the custard over the cooled cake and top with the whipped cream. Garnish with sliced bananas, if desired. Slice and serve.