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 egg yolks with 1/2 cup granulated sugar until thick and pale. Stir in the vanilla extract and mashed bananas until combined.
In a separate bowl, sift together the flour, baking powder, and salt. Gradually fold the dry ingredients into the banana mixture until just combined.
In another clean bowl, beat the egg whites with cream of tartar until soft peaks form. Gradually add the remaining 1/4 cup granulated sugar and continue beating until stiff peaks form.
Gently fold the egg whites into the banana batter in three additions, being careful not to deflate the mixture.
Pour the batter evenly into the prepared pan and spread to the edges.
Bake for 12-15 minutes, or until the cake springs back when lightly touched and a toothpick inserted in the center comes out clean.
While the cake is baking, lay a clean kitchen towel on a flat surface and dust it generously with powdered sugar.
Once the cake is done, immediately loosen the edges with a knife and invert the cake onto the sugared towel. Carefully peel off the parchment paper.
Starting from the short end, roll the cake and towel together into a tight spiral. Allow it to cool completely on a wire rack.
To make the filling, beat the softened cream cheese and butter together until smooth and creamy.
Gradually add the powdered sugar and vanilla extract, beating until well combined and fluffy.
Once the cake is completely cooled, gently unroll it and spread the cream cheese filling 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.