Using a serrated knife, cut cake into ten 1/2-inch slices. Set aside.
For peppermint filling, in a bowl, stir together 1/4 cup of the frosting, 2 to 3 drops peppermint extract, and the red food coloring. Spread five of the cake slices with the filling. Top each with a plain cake slice, making a sandwich. Cut each sandwich into 3 pieces, starting from center of one long side and cutting to each opposite corner, making 15 triangular-shaped pieces; set aside.
For icing, in a small heavy saucepan, cook and stir chopped white chocolate over low heat until melted. Add remaining frosting; heat over low heat, stirring until smooth. Stir in 1/4 teaspoon peppermint extract.
Insert a fork into the side of a cake piece. Holding the cake over the saucepan, spoon on enough icing to cover sides and top. Using another fork, gently push the cake piece onto a wire rack set over a piece of waxed paper. If necessary, spoon icing onto side of cake where fork was inserted. If desired, arrange three peppermint candies on each cake. Repeat with remaining cake pieces, icing, and peppermint candies. Let stand until icing sets.