Preheat the oven to 350°F (175°C). Line a 10×15 inch jelly roll pan with parchment paper and lightly grease it.
In a large bowl, beat the eggs and granulated sugar together with an electric mixer on high speed until thick and pale, about 5 minutes.
Add the vanilla extract and mashed bananas, mixing until combined.
In a separate bowl, whisk together the flour, baking powder, and salt.
Gradually fold the dry ingredients into the banana mixture until just combined.
Gently fold in the vegetable oil until incorporated.
Pour the batter evenly into the prepared pan and spread it out smoothly.
Bake for 12 to 15 minutes, or until the cake springs back when lightly touched.
While the cake is baking, lay a clean kitchen towel on a flat surface and sprinkle it evenly with powdered sugar.
As soon as the cake is done, immediately invert it onto the sugared towel and carefully peel off the parchment paper.
Starting from the short end, roll the cake and towel together into a tight spiral. Let it cool completely on a wire rack.
To make the filling, beat the softened cream cheese and unsalted butter together until smooth and creamy.
Add the powdered sugar and vanilla extract, and beat until fluffy.
Once the cake has cooled, carefully unroll it and spread the cream cheese filling evenly over the surface.
Roll the cake back up without the towel and place it seam side down on a serving plate.
Chill the cake roll in the refrigerator for at least 1 hour before slicing and serving.