Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper. Whisk the flour, baking powder and salt in a medium bowl. Beat the butter and sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Beat in the eggs and peppermint extract until incorporated. Reduce the mixer speed to low and beat in the flour mixture until just combined.
Divide the dough in half and transfer to the prepared baking sheet evenly spaced apart; shape each half into an 8-by-2-inch log. Bake until lightly browned and just firm on the outside, about 25 minutes.
Remove the logs from the oven and let stand 10 minutes; reduce the oven temperature to 325 degrees F. Transfer the logs to a cutting board; slice crosswise on an angle about 1/2 inch thick. Arrange the slices cut-side down on the baking sheet. Return to the oven and bake, flipping halfway through, until dry, about 25 minutes. Transfer to a rack and let cool completely. (The cookies will continue to crisp as they cool.)
Put the white chocolate in a microwave-safe bowl and microwave in 30-second intervals, stirring, until smooth. Dip the biscotti in the white chocolate and sprinkle with the crushed candy canes. Let set, 45 minutes.