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 of granulated sugar until the mixture is thick and pale. Stir in 1 teaspoon vanilla extract and mashed bananas.
In a separate bowl, sift together the all-purpose flour, baking powder, and salt.
Gently 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.
Carefully fold the beaten 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 it out smoothly.
Bake for 12 to 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 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. Allow it to cool completely in this rolled shape.
To prepare the frosting, beat the softened butter until creamy. Gradually add powdered sugar, beating until smooth.
Add 1 teaspoon vanilla extract and 2 tablespoons milk, and beat until light and fluffy.
Unroll the cooled cake gently and spread the frosting evenly over the surface.
Re-roll the cake without the towel. Place the cake roll seam side down on a serving plate.
Chill in the refrigerator for at least 30 minutes before slicing and serving.