Special equipment: a candy cane cookie cutter (about 3 1/2 inches by 2 inches)
For the cookies: In a medium bowl, whisk together the flour, baking powder and salt. In a separate large bowl using an electric mixer, beat together the butter, granulated sugar and brown sugar, mixing until pale yellow and fluffy, 3 minutes. Add in the peppermint extract and eggs and mix well. Slowly incorporate the dry ingredients into the wet ingredients and mix well.
Transfer the dough to a floured work surface, divide into two disks, wrap in plastic and chill in the refrigerator for about an hour.
Preheat the oven to 350 degrees F. Line two baking sheets with parchment paper.
Working with one disk at a time, roll out the dough to 1/4-inch thickness. Cut out candy cane shapes with the cookie cutter and place the cookies about 2 inches apart on the prepared baking sheets. Re-roll the scraps to get as many cookies as possible, returning the dough to the refrigerator to chill if necessary.
Bake just until the edges start to brown, 10 to 12 minutes. (Bake one sheet at a time for even cooking and browning). Transfer the cookies to a wire rack and let cool.
For the royal icing: In the bowl of a stand mixer using a whisk attachment, combine the confectioners' sugar and meringue powder and mix for 30 seconds. Add 1/3 cup water and the peppermint extract and continue mixing on low until the icing forms peaks, 8 to 10 minutes. Adjust the consistency of the icing with additional water if necessary.
Set aside some of the icing in a small bowl and add enough food coloring until the desired shade of red is reached.
Ice the cookies with the white icing and let dry completely. Decorate with the red icing and crushed candies to make a candy cane pattern.