Put the sugar and eggs in a heatproof bowl set over a saucepan of gently simmering water. Whisk, using an electric whisk, for 5–10 minutes until very thick and foamy and the whisk leaves a trail when lifted above the mixture.
Sift over the flour and cocoa powder and carefully fold into the mixture with the melted butter and vanilla until well combined.
Pour the mixture into an oiled and base-lined 20 cm (8 inch) spring-form cake tin and bake in a preheated oven, 180°C (350°F), Gas Mark 4, for 25 minutes until risen and firm to the touch. Remove from the oven and leave to cool in the tin for 5 minutes. Turn out on to a wire rack and leave to cool.
Make the icing. Melt the chocolate and butter together in a heatproof bowl set over a saucepan of gently simmering water. Remove from the heat and beat in the icing sugar. Set aside to cool and then chill for 1 hour until thickened. Beat until pale and fluffy.
Slice the cake in half horizontally and use half of the icing to sandwich the halves back together. Use the remaining icing to cover the top and sides of the cake, swirling the mixture with a palette knife.