Preheat the oven to 350°F (175°C). Line the baking sheet with parchment paper.
In a mixing bowl, beat the eggs and granulated sugar with an electric mixer on high speed for about 5 minutes, until light and fluffy.
In another bowl, sift together the flour, cocoa powder, baking powder, and salt. Gradually fold this dry mixture into the egg mixture using a rubber spatula.
Carefully fold in the milk and vegetable oil until just combined.
Pour the batter onto the prepared baking sheet and spread evenly. Bake in the preheated oven for 12-15 minutes or until a toothpick inserted in the center comes out clean.
While the cake is baking, prepare a clean kitchen towel dusted with powdered sugar. Once the cake is done, immediately invert it onto the towel. Peel off the parchment paper and roll the cake with the towel starting from the short edge. Let it cool completely in this rolled form.
In a separate bowl, whip the heavy cream until soft peaks form. Add in the powdered sugar and vanilla extract and continue to whip until stiff peaks form. Gently fold in the mashed bananas.
Once the cake has cooled, carefully unroll it. Spread the banana cream mixture evenly over the surface, then drizzle the melted chocolate chips over the cream.
Roll the cake back up without the towel, and wrap it tightly in plastic wrap. Refrigerate for at least 1 hour to set.
When ready to serve, unwrap the roll and slice it into pieces. Enjoy!