Preheat the oven to 350°F (175°C).

Prepare a 15x10-inch jelly roll pan by greasing it and lining it with parchment paper.

In a mixing bowl, mash the ripe bananas thoroughly until smooth.

In a separate bowl, beat the eggs and sugar together until light and frothy, about 5 minutes.

Gently fold in the mashed bananas and vanilla extract into the egg mixture.

Sift together the flour, baking powder, and salt, then fold them into the wet ingredients until just combined.

Pour the batter into the prepared pan, spreading it evenly.

Bake in the preheated oven for 12-15 minutes, or until the cake is lightly golden and springs back when touched.

Remove from the oven and cool for 5 minutes, then invert the cake onto a clean kitchen towel dusted with powdered sugar.

Roll the cake up gently in the towel, starting from one short end, and allow it to cool completely in that position.

While the cake cools, whip the heavy cream with powdered sugar until stiff peaks form.

Unroll the cooled cake and spread the whipped cream filling evenly over the surface.

Re-roll the cake carefully without the towel and place it seam-side down on a serving platter.

Chill the rolled cake in the refrigerator for at least one hour before slicing.

Slice and serve the banana roll cake, optionally dusting with powdered sugar before serving.