Beat first 3 ingredients at medium speed with an electric mixer until creamy. Beat in extracts. Gradually add powdered sugar alternately with 2 Tbsp. milk. Beat at low speed until blended and smooth after each addition. Add up to 1 Tbsp. milk, 1 tsp. at a time, beating until frosting reaches desired consistency.
Reserve 2 cups frosting in a small bowl. Process peppermint candies in a food processor until finely crushed. Stir 1/4 cup crushed candies into reserved 2 cups frosting.
Place 1 Basic White Cake layer on a serving platter. Spread 1 cup peppermint candy frosting mixture over cake layer. Top with second layer, and spread remaining 1 cup peppermint candy frosting over cake layer. Top with third cake layer, and spread top and sides of cake with remaining plain peppermint frosting. Sprinkle with remaining crushed candies, or top with meringues or fondant bow.
For the cover look, buy 1 lb. each of red and white rolled fondant from the cake section of a crafts store. Roll a thin red strip, and follow our steps below for the bow. Dust your countertop with cornstarch or powdered sugar to prevent sticking.
Make the Bow: Create a Cover-Worthy Cake Garnish
Roll Long Strips: You'll need a rolling pin, pizza wheel, ruler, and red and white fondant. Roll a chunk of red and chunk of white fondant into two long strips (about 1/8 inch thick). Using ruler and pizza wheel, cut white fondant into thin strips.
Make Striped Ribbon: Moisten finger with water; rub one side of one thin white strip. Place strip, moist side down, on red to form a stripe. Repeat with two to three strips. Roll gently with rolling pin. Repeat as needed to make more ribbons.
Cut, Shape, Dry: Cut two identical ribbons for tails. Place foil under tails to create folds. Form loops of bow by wrapping ribbon around a cardboard roll, leaving 1/4 inch excess on ends. Pinch ends. Cut a small rectangle for knot. Let dry 24 hours.
Assemble Bow: Place bow tails on cake, draping down sides. Place bow loops over ribbon on top of cake (loop ends touching). Tip loops at an angle, and press gently into frosting to secure. Cover seam with knot, using frosting as glue, if needed.