Whisk flour, baking soda, and salt in a bowl. Beat butter, vegetable shortening, light brown sugar, and white sugar in a separate large bowl until creamy. Beat eggs, vanilla and peppermint extracts, and red food coloring into butter mixture until smooth. Gradually beat dry ingredients into wet ingredients until dough is smooth. Fold white chocolate chips and crushed candy canes into dough.
Pinch off 1-tablespoon-sized pieces of dough, roll into balls, and place on ungreased baking sheets. Lightly flatten cookies with the bottom of a drinking glass.
Bake in the preheated oven until cookies are set but not browned, 9 to 10 minutes. Let cool on baking sheets for 2 minutes before removing to finish cooling on wire racks.