Combine sugar, syrup, and water in a medium-size heavy saucepan. Cook over medium heat, stirring frequently, until mixture comes to a boil and sugar is dissolved. Continue cooking, stirring frequently, until mixture reaches soft ball stage (240°).
Beat egg whites (at room temperature) until foamy. While beating at medium speed of electric mixer, slowly pour half of the hot syrup in a thin stream over whites. Return remaining syrup to heat, and continue cooking until mixture reaches hard ball stage (260°).
Continue beating at medium speed of electric mixer, and slowly pour remaining syrup in a thin stream over egg whites. Turn mixer to high speed, and continue beating until stiff peaks form and frosting is thick enough to spread. Add peppermint oil; beat until blended. Spread immediately on cooled cake. Drizzle melted chocolate over frosting.