Heat the oven to 375°F. Grease and line a 15½ × 10½-inch jelly-roll pan, then grease the paper.
Combine the flour, cornstarch, cocoa, baking powder and salt. Separate the eggs into two mixing bowls.
Whip the whites with ½ cup of the sugar until they hold firm peaks. In the other bowl beat the yolks, remaining sugar and vanilla until pale.
Mix the dry ingredients into the yolks. Loosen this mixture with a quarter of the whites, then fold in the rest without knocking out the air.
Spread into the pan and bake for 12–14 minutes, until the surface springs back. Wait 2 minutes, turn onto a generously sugared clean towel and remove the paper.
Roll the warm sponge and towel together from a short edge. Let it cool completely before unrolling.
Spread with softened peppermint ice cream. Roll up again without the towel, wrap securely and freeze overnight.
Gently melt the cream, chocolate and corn syrup together. Stir in the vanilla off the heat, then let the glaze reach room temperature.
Coat the frozen roll with the glaze and scatter crushed candy canes over it. Freeze for about an hour before slicing.