Make the cake: Position racks in the middle and lower thirds of the oven and preheat to 350 degrees F. Lightly butter three 8-inch-round cake pans and line the bottoms with parchment paper; butter the parchment and dust with flour, tapping out the excess. Sift the flour, baking powder and salt into a medium bowl. Combine the milk and the vanilla and peppermint extracts in a small bowl; set aside.
Beat the butter in a large bowl with a mixer on medium speed until smooth, about 1 minute. Gradually beat in 1 cup sugar, then increase the mixer speed to high and continue beating until light and fluffy, about 5 minutes.
Reduce the mixer speed to low and add the flour mixture in 3 additions, alternating with the milk mixture, beginning and ending with the flour mixture, until just combined. Scrape down the side of the bowl, then increase the mixer speed to medium and beat until smooth, about 5 more minutes.
In a separate large bowl (using clean beaters), beat the egg whites with a mixer on high speed until foamy. Gradually add the remaining 1/2 cup sugar and continue beating until stiff, shiny peaks form, about 6 minutes. Using a rubber spatula, gently fold the whites into the batter in 3 additions until combined.
Divide the batter evenly among the prepared pans. Bake, switching the position of the pans about halfway through, until a toothpick inserted into the center comes out clean, about 20 minutes. Let the cakes cool 10 minutes in the pans, then run a knife around the edges and invert onto racks to cool completely. Peel off the parchment. (The cakes can be made up to 1 day ahead; wrap in plastic and store at room temperature.)
Make the frosting: Fill a medium saucepan with 1 to 2 inches of water and bring to a simmer over medium-high heat. Combine the sugar, corn syrup, egg whites, 2 tablespoons water and the salt in a heatproof bowl and set over the saucepan (do not let the bowl touch the water). Whisk until the sugar dissolves and the mixture registers 165 degrees F on a candy thermometer, about 8 minutes. Remove the bowl from the pan and beat with a mixer on medium speed until soft peaks form, about 5 minutes. Increase the mixer speed to high and beat until fluffy and cool, about 4 more minutes. Add the vanilla and peppermint extracts and beat until combined, 1 more minute.
Assemble the cake: Place 1 cake layer on a platter and spread 1 cup frosting on top; sprinkle with about 2 tablespoons crushed candy canes. Top with another cake layer, spread with 1 cup frosting and sprinkle with another 2 tablespoons crushed candy canes. Top with the final cake layer, then frost the top and side of the cake with the remaining frosting. Sprinkle more crushed candy canes on top.