Preheat oven to 325°F. Grease and line an 8x8-inch baking pan with parchment paper.
In a large mixing bowl, beat the egg whites with a pinch of salt until stiff peaks form. Set aside.
In another bowl, beat the egg yolks, granulated sugar, and brown sugar until pale and creamy.
Mix in the melted butter and mashed bananas until combined.
Sift in the flour and mix just until incorporated.
Slowly whisk in the lukewarm milk and vanilla extract until smooth (batter will be thin).
Gently fold in the beaten egg whites in three parts, keeping some fluffy lumps – do not overmix.
Pour the batter into the prepared pan and bake for 45–55 minutes, until the top is golden and set.
Cool completely, dust with powdered sugar, slice, and serve.