Preheat the oven to 325°. Butter a 9-inch round cake pan and line it with parchment paper; butter the parchment paper. Dust the pan all over with flour, tapping out any excess. In a medium bowl, whisk the flour with the baking powder, baking soda and salt.
In a large bowl, using a handheld electric mixer, beat the butter and sugar at medium-high speed until light and fluffy, about 3 minutes. Beat in the egg yolks and vanilla extract. Add the mashed bananas and beat the mixture until smooth. Add half of the dry ingredients and beat at low speed until the batter is moistened. Beat in half of the buttermilk, then add the remaining dry ingredients and the remaining buttermilk.
In a medium bowl, beat the egg whites at medium-high speed until firm peaks form. Beat one-fourth of the egg whites into the batter at low speed. Using a rubber spatula, fold in the rest of the whites until no streaks remain. Scrape the batter into the prepared pan and smooth the top. Bake the cake for about 40 minutes, until the top is golden and springy and a toothpick inserted in the center comes out clean; the top will be very slightly cracked. Let cool in the pan for 15 minutes, then invert the cake onto a rack to cool completely. Peel off the parchment paper.
In a medium bowl, beat the mascarpone with the confectioners’ sugar at medium speed until the frosting is light and fluffy, about 2 minutes. Using a large serrated knife, cut the cake horizontally into 3 layers. Place the top layer, cut side up on a cake plate and spread with one-third of the mascarpone frosting. Arrange half of the bananas in a single layer on the frosting. Top with the middle cake layer. Cover the layer with another third of the frosting and top with the remaining bananas. Cover with the bottom cake layer, cut side down, and frost the top. Refrigerate the cake for at least 30 minutes before serving.
The baked and cooled cake can be tightly wrapped in plastic and refrigerated for 3 days or frozen for up to 1 month. The assembled cake can be made earlier in the day and refrigerated.
Cooking Club Tip: Stick several toothpicks around the circumference where you intend to separate the cake into layers. Then with a serrated knife or piece of taut, unwaxed, unflavored dental floss, slice the cake horizontally (using the toothpicks as a guide), gently sawing back and forth to cut.