Preheat the oven to 350 degrees F. Line a 10x15-inch jelly roll pan with parchment paper and lightly grease the paper.

Beat the eggs and granulated sugar together for 3 to 4 minutes, until thick and pale. Stir in the mashed bananas and vanilla extract.

Whisk the flour, baking powder, and salt together in a separate bowl. Gently fold the dry ingredients into the banana mixture until just combined.

Spread the batter evenly in the prepared pan. Bake for 12 to 14 minutes, until the cake springs back when lightly touched.

While the cake bakes, lay a clean kitchen towel on the counter. Dust it generously with powdered sugar.

Turn the cake out onto the towel as soon as it comes out of the oven. Carefully peel off the parchment paper.

Starting at one short end, roll the warm cake up inside the towel. Let it cool completely while rolled, about 1 hour.

Beat the cream cheese and butter together until smooth. Add the powdered sugar and vanilla, then mix until creamy and spreadable.

Unroll the cooled cake gently. Spread the filling evenly over the surface, leaving a small border around the edges.

Re-roll the cake without the towel, as tightly as you can. Wrap in plastic wrap and refrigerate at least 1 hour.

Dust with more powdered sugar before serving if you like. Slice with a sharp knife and serve chilled.