Preheat oven to 350°F. Line two baking sheets with parchment paper.
Whisk flour, baking powder, and salt in a bowl. Set aside.
Cream butter and sugar until very fluffy, about 2 minutes.
Beat in the egg, vanilla, and peppermint until smooth.
Mix the dry ingredients into the butter mixture just until a soft dough forms.
Divide dough in half. Tint one half red with food coloring; leave the other half plain.
Roll each half between parchment into a 9x12-inch rectangle. Chill 10 minutes to firm.
Stack red dough over plain dough. Trim edges for clean sides, then roll tightly from a long edge into a log.
Roll the log in crushed candy canes to coat the outside. Wrap and chill 30 minutes.
Slice into 1/4-inch rounds. Arrange 2 inches apart on prepared sheets.
Bake 9–11 minutes, until set and just barely golden at the edges. Cool on a rack.