Preheat the oven to 180 C / Gas 4. Butter and flour a 20cm square cake tin.
Whisk the eggs with the icing sugar. Sieve the flour and stir in the flour and melted butter and whisk until smooth. Stir in the mashed bananas and chocolate pieces. Pour cake mixture into the prepared cake tin.
Bake in the preheated oven until a skewer inserted near the centre comes out clean, about 35 minutes. Cool for 10 minutes in the cake tin, then turn the cake out onto a wire rack to cool completely.