Let butter and eggs stand at room temperature for 30 minutes. Lightly grease bottoms of two 8x8x2-inch square cake pans. Line bottoms of pans with waxed paper. Grease and lightly flour bottoms and sides of pans; set aside. In a medium bowl stir together flour, cocoa powder, baking soda, baking powder, and salt; set aside.
Preheat oven to 350 degrees F. In a large mixing bowl beat butter with an electric mixer on medium to high speed for 30 seconds. Gradually add sugar, about 1/4 cup at a time, beating on medium speed until well combined and scraping sides of bowl. Beat on medium speed for 2 minutes more. Add eggs, one at a time, beating until combined after each addition. Beat in vanilla. Alternately add flour mixture and milk to butter mixture, beating on low speed after each addition just until combined. Beat on medium to high speed for 20 seconds more.
Spread batter evenly into prepared pan(s). Bake in the preheated oven for 35 to 40 minutes or until a toothpick inserted near the centers comes out clean. Cool cake layers in pans on wire racks for 10 minutes. Remove cake layers from pans; peel off waxed paper. Cool thoroughly on racks.
To assemble, place a cake layer on a serving plate. Stir the 1/3 cup finely chopped peppermint sticks into the Peppermint Fudge Filling. Spread the filling evenly over the cake. Top with the remaining cake layer. Spread Chocolate Whipped Cream Frosting over top and sides of cake. Sprinkle chocolate curls and coarsely chopped peppermint sticks in a wide stripe pattern over top of cake.* Cover and chill for up to 4 hours. Store any leftover cake in the refrigerator. Makes 12 to 16 servings
Peppermint Fudge Filling
In a small saucepan heat whipping cream over medium heat until just simmering. Remove from heat. Add chopped bittersweet chocolate; let stand, uncovered, for 5 minutes. Stir mixture until smooth. Stir peppermint extract into chocolate mixture in bowl. Let stand 5 to 10 minutes or until mixture thickens. Stir in chopped peppermint sticks, candy canes, or other hard peppermint candies.
Chocolate Butter Frosting
In a large mixing bowl beat butter with an electric mixer on medium speed until smooth. Gradually add 2 cups of the powdered sugar and unsweetened cocoa powder, beating well. Beat in 1/4 cup milk and vanilla. Gradually beat in 3 cups additional powdered sugar and 2 tablespoons additional milk until a fluffy spreadable consistency is reached.