Preheat the oven to 350°F (175°C). Line a 15×10-inch jelly roll pan with parchment paper and lightly grease it.
In a large mixing bowl, beat the eggs with an electric mixer on high speed until thick and pale, about 3-4 minutes.
Gradually add the granulated sugar while continuing to beat until the mixture is fluffy.
Add vanilla extract and mashed bananas to the egg mixture and mix until combined.
In a separate bowl, whisk together the all-purpose flour, baking powder, and salt.
Gently fold the dry ingredients into the banana mixture using a spatula until just combined.
Fold in the melted and cooled butter until fully incorporated.
Pour the batter evenly into the prepared jelly roll pan and spread it out smoothly.
Bake for 12-15 minutes or until the cake springs back when lightly touched and a toothpick inserted comes out clean.
While the cake is baking, lay a clean kitchen towel on a flat surface and sprinkle it evenly with powdered sugar.
Once the cake is done, immediately invert it onto the prepared towel. Carefully peel off the parchment paper.
Starting from the short edge, roll the cake and towel together into a tight spiral. Let it cool completely on a wire rack.
To make the frosting, beat the softened cream cheese and butter together until smooth and creamy.
Gradually add powdered sugar and vanilla extract, beating until the frosting is light and fluffy.
Unroll the cooled cake gently and spread the cream cheese frosting evenly over the surface.
Roll the cake back up without the towel, wrap it in plastic wrap, and refrigerate for at least 1 hour before serving.
Slice and serve chilled or at room temperature.