1) Preheat the oven to 375°F. Prepare a jelly roll pan by lining with parchment and spritzing with cooking spray.
2) After separating the eggs, place the whites in the bowl of your stand mixer and whip on high speed to soft peaks. Gradually add 1/4 cup of the granulated sugar, continue to beat until you reach stiff but not dry peaks. Set aside in a clean bowl.
3) To make the cake batter: Using a stand mixer, whip the egg yolks until thick and pale yellow, about 5 minutes on high speed. Gradually beat in the remaining 1/2 cup sugar.
4) In a separate small bowl combine the flour, salt, baking powder, and baking soda. Whisk until well combined and lump free.
5) Slowly stir the mashed banana into the egg yolk mixture, followed by the flour mixture.
6) Gently fold in the beaten egg whites, stirring until well combined. Take care not to deflate the batter as you fold.
7) Pour the cake batter into the prepared pan and bake for 8 to 12 minutes. The edges will begin to pull away from the sides of the pan, and the center will spring back when lightly pressed.
8) Remove the cake from the oven, and lift it out of the pan by lifting the parchment paper. Place the cake on the counter, and trim off its edges to make rolling easier. Turn the cake over and peel off the parchment paper.
9) Using a large, clean kitchen towel well dusted with confectioners' sugar, roll the cake up as though the towel were the filling. Cool the cake, rolled up, for 20 to 30 minutes.
10) While the cake is cooling, prepare the filling. To make the rum syrup, place the brown sugar and water in a medium-sized sauté pan over medium heat. Cook until the sugar dissolves and the liquid turns syrupy, stirring occasionally. Remove from the heat and stir in the butter and rum. Allow to cool to room temperature. If you prefer, you can leave out the rum and flavor the syrup with 1/4 teaspoon of rum extract instead.
11) Beat the softened cream cheese and confectioners' sugar in a bowl until the mixture is lump-free and fluffy. Add a pinch of salt, plus 1/4 cup of the prepared rum syrup.
12) To assemble the cake: Unroll the towel and remove it. Lightly brush the cake with 2 to 3 tablespoons of the rum syrup. Spread the cake with the filling, and re-roll. Place in the refrigerator to chill for 20 to 30 minutes. The roll can be frozen at this point for up to 3 months.
13) To serve, slice with a sharp serrated knife and plate with an extra drizzle of rum syrup. Refrigerate any leftover cake or syrup.
Yield: 10 servings.