Preheat the oven to 350°F (175°C). Line a 10×15-inch jelly roll pan with parchment paper and lightly grease the paper.
In a large bowl, beat the egg yolks with 1/2 cup of granulated sugar until the mixture is thick and pale. Stir in the vanilla extract and mashed bananas.
In a separate bowl, whisk together the all-purpose 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 jelly roll pan and spread it out smoothly.
Bake for 12 to 15 minutes, or until the cake springs back lightly when 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 sprinkle it lightly 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 while rolled to set its shape.
To prepare the filling, beat the softened cream cheese and butter together until smooth and creamy.
Add the powdered sugar and vanilla extract to the cream cheese mixture and beat until fully combined and fluffy.
Once the cake has cooled, gently unroll it and spread the cream cheese filling evenly over the surface.
Roll the cake back up without the towel, keeping it tight but gentle to avoid cracking.
Wrap the cake roll in plastic wrap and refrigerate for at least 1 hour before slicing and serving.