In a bowl of stand mixer, cream together the butter and shortening. Add sugars and cream until light and fluffy, 1 to 2 minutes. Add eggs, incorporating one at a time. Mix in vanilla and peppermint extracts.
In a large bowl, combine the cornstarch, cocoa powder, espresso powder, flour, salt and baking soda. Add the dry mixture to the creamed mixture and mix until just combined. Stir in 1 cup of mini chocolate chips.
Line a large cookie sheet with parchment paper. Using an ice cream scoop, scoop out cookies and place on prepare cookie sheet (they can be close together as they are not being baked right away). Press down slightly then sprinkle each with the remaining chocolate chips and crushed candy canes. Lightly press to make sure the toppings stick. Place in the freezer for 1 to 2 hours.
When ready to bake, preheat oven to 350 degrees F. Bake 3 inches apart on parchment lined cookies sheets for 15 to 18 minutes. Allow to cool on pan for 2 minutes then move to a cooling rack to cool completely.
Make ahead freezer option: Once the cookie dough has frozen for 2 hours, place cookies into a freezer safe bag. Bake cookies as needed according to the directions above.