Preheat the oven to 350°F (175°C). Line a 15×10-inch jelly roll pan with parchment paper and lightly grease the paper.
In a large bowl, whisk the egg yolks with 1/4 cup granulated sugar until thick and pale. Stir in 1 teaspoon 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 a clean bowl, beat the egg whites and cream of tartar with an electric mixer until soft peaks form. Gradually add the remaining 1/4 cup granulated sugar and continue beating until stiff peaks form.
Gently 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 jelly roll 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 dust it generously with powdered sugar.
Once the cake is done, immediately loosen the edges with a knife and invert the cake onto the prepared towel. 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 frosting, beat the softened butter until creamy. Gradually add the powdered sugar, beating until smooth.
Add vanilla extract and milk, then beat until light and fluffy.
Once the cake has cooled, gently unroll it and spread the frosting evenly over the surface.
Re-roll the cake without the towel and place it seam side down on a serving plate.
Chill the cake roll for at least 30 minutes before slicing and serving.