Preheat your oven to 350°F (175°C) and prepare your cookie sheets by lining them with parchment paper to prevent sticking.

In a large mixing bowl, cream together the softened butter, shortening, white sugar, and brown sugar. Beat on medium speed for about two minutes until the mixture becomes light and fluffy.

Add the vanilla extract and peppermint extract, blending them into the creamed mixture. Incorporate the eggs one at a time, ensuring each is fully mixed before adding the next.

Lower the mixer speed to its minimum setting and gradually sift in the flour, baking soda, and salt. Mix until a soft, pliable dough forms. If the dough feels too sticky when pressed between your fingers, add a tablespoon of flour at a time until the desired consistency is achieved.

Gently fold in the crushed candy canes and white chocolate chips, ensuring they are evenly distributed throughout the dough.

Portion the dough into small balls, approximately 1 to 1.5 tablespoons each, and place them on the prepared cookie sheets, ensuring at least two inches of space between each ball to allow for spreading.

Bake the cookies in the preheated oven for 8 to 10 minutes, or until the tops appear just set. Remove the trays from the oven and immediately dot the tops of the cookies with additional white chocolate chips if desired for a decorative touch.

Allow the cookies to cool on the baking sheets for 10 minutes to firm up slightly. Transfer them to a wire rack to cool completely before serving or storing.