Wash your hands. Preheat oven to 350 degrees, and spray two 8” round cake pans with non-stick cooking spray.
Whisk eggs in a large bowl, then add 10 tablespoons of melted butter and stir until mixture thickens. Next, add the peppermint extract and continue whisking.
Add 1/3 of the vanilla cake mix to the egg/butter mixture and whisk gently until blended. Gently whisk ½ of the yogurt into the batter, followed by another 1/3 of the dry mix. Add the remaining yogurt and dry mix, and whisk until smooth.
Evenly distribute batter into the cake pans and bake for 18-20 minutes or until the top of the cakes are golden. Allow cakes to cool before removing from pans.
While the cakes are in the oven, use a rolling pin (or some other heavy tool) to crush the candy canes into small pieces.
Use an electric hand mixer to cream the remaining 6 tablespoons of butter and 4 tablespoons of yogurt in a large bowl. Slowly add the vanilla frosting mix and continue whisking at medium speed until frosting is smooth and all lumps are gone.
To assemble the cake, place the first layer on a cake stand and spread a thin layer of frosting over the entire surface. Top with the second layer of cake. Spread frosting all over the top and sides of the cake. Apply crushed candy canes to the sides of the cake until you can no longer see the frosting. Garnish the top with a few more sprinkles of crushed candy.