Preheat the oven to 350º F. Line a large baking sheet (17x12-inches) with parchment paper.
In a medium bowl, whisk together the flour, baking powder, and salt; set aside. In a large bowl, cream the butter and sugar until light and fluffy.
Scrape down the sides of the bowl and add the eggs one at a time, beating well after each addition.
Scrape down the sides of the bowl again and add the vanilla and peppermint extract.
Add the flour; mix until flour is completely incorporated.
Fold in the finely crushed candy canes.
Transfer the dough to a lightly floured work surface, dividing it in half. Form each piece of dough into a 16x1½-inch log (you may adapt the size of the log to a smaller pan).
Place the logs on the prepared sheet pan about 4-inches apart; put baking sheet in the freezer for 10 minutes for the dough to chill.
Bake for 25-30 minutes or until lightly golden and still slightly soft on the inside. Remove the pan from the oven and allow the logs to rest on the pan for 10-15 minutes.
Reduce the oven temperature to 325º F.
Carefully transfer the logs to a cutting surface. Using a sharp or serrated knife, cut the logs into ¾-inch wide slices on the diagonal. Return the slices cut side down to baking sheet.
Bake for 20 minutes. Remove from the oven and cool completely on the baking sheets.