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Heat the oven to 180°C/160°C fan/gas Using an electric mixer, beat the eggs and sugar in a large mixing bowl for about 10 minutes until doubled in size. Fold in the flour, cocoa, espresso powder and milk with a large metal spoon, pour into the prepared tin and level with a spatula. Bake for 10 minutes until risen and springy.
Lay a damp tea towel on the work surface and cover with a large sheet of non-stick baking paper. Dust the paper liberally with icing sugar then, in one swift movement, carefully and confidently turn the roulade out onto the sheet. Peel away the baking paper from the base of the sponge.
Using a sharp knife and being careful not to cut all the way through, score a line all the way along one short edge of the sponge, 1cm from the edge. Starting from this edge, roll the sponge tightly, using the paper and tea towel to help. Wrap in the tea towel and leave to cool completely.
For the filling, whisk the cream, Kahlúa, vodka and icing sugar in a bowl until just holding its shape.
Carefully unroll the cooled roulade and spread over the cream, leaving a 2cm border. Re-roll and dust with icing sugar. Decorate with chocolate curls and slice to serve.