Preheat oven to 350 degrees F (175 degrees C). Line two baking sheets with parchment paper.
Beat white sugar, butter, and cocoa powder together with an electric mixer in a large bowl until creamy and smooth. Add eggs, one at a time, beating well after each addition. Mix chocolate liqueur and peppermint extract into the sugar-egg mixture.
Combine flour, baking powder, and salt in a separate bowl. Slowly mix flour mixture into sugar-egg mixture until fully incorporated; fold mint chocolate chips into the dough.
Divide dough into 4 equal parts and shape into logs. Arrange the logs on the prepared baking sheets.
Dip a spatula in water and use it to smooth surface of the logs.
Bake biscotti logs in the preheated oven until firm to the touch, 30 to 35 minutes. Cool biscotti logs completely on wire racks.
Reduce oven temperature to 300 degrees F (150 degrees C).
Cut biscotti logs into 3/4-inch slices and arrange on baking sheets.
Bake in the oven until biscotti are dry, about 10 minutes per side. Cool completely on wire racks.
Place candy melts in a wide microwave-safe bowl; heat in microwave until melted, about 2 1/2 minutes, stirring every 30 seconds.
Dip cooled biscotti in the melted white candy; sprinkle with crushed candy canes.