Preheat oven to 350F. Lightly grease a 9x13 baking pan. Set aside. In a large bowl, beat butter and sugar together until completely creamy. Beat egg in one at a time until smooth, then add milk and vanilla.
In a separate bowl, combine flour, baking powder, and salt. Fold into liquid mixture until batter is just combined. Scrape batter into prepared pan and set aside.
Bake sheet cake at 350F 20-25 minutes or until golden brown and a toothpick tests done. Cool completely on a wire cooling rack.
Pudding
In a medium saucepan, bring milk and sugar to a low simmer over medium heat. In a separate bowl, whisk cornstarch and egg yolks and gradually whisk in a little of hot milk mixture until smooth.
Add all of egg yolk mixture to saucepan and whisk frequently over medium heat until pudding is bubbly and thickened. Remove from heat and whisk in vanilla. Cool completely.
Once vanilla pudding is cooled, gently fold in whipped cream. Spread pudding mixture over cooled cake and top with bananas. Chill cake until ready to serve. Enjoy!