Preheat the oven to 350 degrees F and line three baking sheets with parchment paper. Using a stand mixer fitted with the paddle attachment (or a large mixing bowl with a hand mixer), combine the butter, sugars, eggs and vanilla extract and beat on medium speed until creamy and free of lumps. Add Nutella and beat on low speed until just combined.
In a separate bowl, combine the flour, cocoa powder, baking soda, baking powder and salt. Mix well. Gradually add to the butter-sugar mixture and beat on medium speed until just combined. Stir in the chocolate chips. Using a medium cookie scoop, place 1.5-tablespoon-size chunks of batter in your hands and roll into balls. Place onto the prepared baking sheets and bake in the preheated oven for 10-12 minutes. Let cool.
MAKE THE FROSTING
Using a stand mixer fitted with the whisk attachment (or a large mixing bowl with a hand mixer), combine the confectioner’s sugar, butter and peppermint extract. Beat on medium speed until moist. Scrape the sides of the bowl if necessary. Add the milk and beat on medium-high speed for 3 minutes.
ASSEMBLE THE COOKIES
Spread 2 teaspoons of frosting over the top of each cooled cookie. Top with candy cane pieces.