Preheat oven to 350F (175C) and line cookie sheets with parchment paper. Set aside.
In a large, microwave safe bowl, combine butter and 2 cups semisweet chocolate chips. Microwave for 30 seconds and stir, repeating at 15 second increments until chocolate and butter are completely melted.
Remove from microwave and allow to cool at least 10 minutes or until mixture no longer feels warm to the touch before proceeding.
Once butter/chocolate mixture has cooled, stir in sugars, egg, egg yolk, and vanilla and peppermint extracts.
In a separate, medium-sized bowl, whisk together flour, cocoa powder, cornstarch, baking powder, baking soda, and salt.
Gradually stir flour mixture into butter mixture until completely combined.
Stir in mini chocolate chips, if using.
Drop cookie dough by 1 ½ Tablespoon scoop onto parchment paper, pressing down slightly to flatten.
Bake 9 minutes, allow to cool on cookie sheet for 5 minutes and then transfer to cookie rack to cool completely before dipping in chocolate.
Prepare white chocolate by placing in medium-sized microwave-safe bowl with shortening or coconut oil. Microwave for 30 seconds and stir, then continue to microwave in 15-second increments (stirring well in-between) until chocolate is completely melted.
Dip cookies ⅓-½ of the way in white chocolate and place on a wax paper lined baking sheet. Sprinkle with crushed candy canes. Allow chocolate to harden before enjoying.