Prepare Classic Génoise as directed in a 9-inch springform pan. Cool the cake completely.
To assemble cake, split génoise into 3 layers horizontally using a serrated knife. Place bottom layer, cut side up, in a 9-inch springform pan; place layer in pan and remaining layers in freezer.
Place a large bowl in freezer. Remove the ice cream from freezer, and let stand at room temperature until slightly soft. Spoon ice cream into chilled bowl. Stir in crème de menthe and mint thins until well combined. Spoon half of ice cream mixture onto cake layer in pan; top with middle layer of cake. Repeat procedure, ending with top layer of cake, cut side down. Return cake to freezer.
To prepare frosting, combine sugar and 3 tablespoons cocoa in a small bowl. Combine milk and butter in a 1-cup glass measure; microwave at HIGH 30 seconds or until butter melts. Add milk mixture and vanilla to sugar mixture, stirring with a whisk until smooth. Spread frosting over top of cake, and freeze for 30 minutes. Cover with plastic wrap, and freeze 4 hours or until firm.
Place cake in refrigerator 30 minutes before serving to soften. Remove band from springform pan. Sprinkle 1 teaspoon cocoa over cake.