Prepare the Cake Layers: Preheat oven to 350°F. Beat granulated sugar and butter with an electric mixer on medium speed until light and fluffy, about 2 minutes. Add eggs, 1 at a time, and beat just until combined after each addition. Add vanilla, and beat just until combined. Whisk together flour, baking powder, baking soda, and salt in a medium bowl. Add flour mixture, in batches, alternately with milk, beginning and ending with flour mixture. Beat on low speed just until blended after each addition. Pour batter into 2 greased (with shortening) and floured 9-inch round cake pans.
Bake in preheated oven until a wooden pick inserted in center comes out clean, 28 to 30 minutes. Cool in pans on wire racks 20 minutes; remove from pans to wire racks, and cool completely, about 30 minutes.
Prepare the Peppermint Buttercream: Beat powdered sugar and butter with electric mixer on medium speed until smooth, about 2 minutes. Add crushed peppermints, vanilla, and salt; beat until blended. Beat in 1 teaspoon of the milk. Beat in up to 1 more teaspoon of milk, 1/4 teaspoon at a time, until desired consistency is reached. Beat in food coloring, 1 drop at a time, until desired color is reached.
Prepare the Seven-Minute Frosting: Pour water to a depth of 2 inches into bottom of a saucepan or double boiler; bring to a simmer over medium heat. Stir together egg whites, granulated sugar, corn syrup, and salt in a heatproof bowl or top of double boiler. Place bowl over simmering water, and whisk constantly until sugar dissolves and mixture is hot, about 7 minutes. Remove from heat, and beat with an electric mixer on medium-high until stiff peaks form and mixture is completely cooled, about 10 minutes. Beat in peppermint extract.
Assemble Cake: Place 1 Cake Layer on a cake plate. Spread Peppermint Buttercream in an even layer, about 1/2 inch thick, to within 1/2 inch of cake edge. Top with remaining Cake Layer.
Spread cooled Seven-Minute Frosting over top and sides of cake. Reserve remaining Frosting for garnish (see Garnish How-to, below).
Prepare the candy: Roughly crush 1 cup hard peppermint candies into small pieces. Crush an additional 1/4 cup candies into fine pieces and set aside. Add 1 tsp. white Disco Dust to 1 cup crushed candy, and stir to combine.
Decorate the cake: Spoon the remaining Seven-Minute Frosting into a ziplock plastic bag, and snip 1 corner of the bag. Pipe swirls of frosting on the top of the cake. Lightly press the Disco Dust-coated crushed candy into the sides of the frosted cake until the sides of the cake are coated. Sprinkle the piped swirls with the finely crushed candy and an additional 1/2 tsp. Disco Dust.