Place chocolate in a medium bowl. In a small saucepan, bring heavy cream to a boil over high heat; stir in peppermint extract. Strain cream into chocolate. Let stand 5 minutes; stir until smooth (mixture will appear oily at first.) Cool until mixture falls in ribbons when spoon is lifted, 2 to 6 minutes.

To ice cake, set the cake on a serving platter. Tuck parchment or waxed paper under sides of cake (to keep platter clean). Pour icing onto center of cake. Use an offset spatula or a dinner knife to spread evenly over the top and down the sides. Let set, about 30 minutes.