Whisk eggs for about 3 minutes until thick and foamy. Add sugar and vanilla and whisk together.
Sift flour, cocoa powder and baking powder into a medium sized bowl and fold gently into the egg mixture until just combined.
Line a 12" x 8" sponge tin with baking paper and pour the batter in. It will be quite sticky so use a rubber spatula to spread it to the edges.
Bake for 8-10 minutes at 180 degrees C until a toothpick inserted into the centre comes out clean.
While the cake is baking, lay out a clean tea town and cover an area a little bit bigger than the size of the cake in icing sugar.
As soon as you pull the cake out of the oven, turn it out of the tin onto the part of the tea towel covered in sugar. Roll the cake up tightly in the tea towel from the long side, leaving the tea towel wrapped in the cake. Leave the cake to cool completely (1-2 hours) and then gently unwrap, removing the tea towel.
Spread with jam and whipped cream, and roll the cake up again.
To make the icing sift the icing sugar and cocoa powder into a bowl and add the milk, 1 tablespoon at a time, until the icing is smooth and spreadable.
Ice the cake, sprinkle with icing sugar and leave to set for 30 minutes before slicing and serving.