Sift flour, baking powder, baking soda, and salt into a large mixing bowl.
Cream butter and sugar in another large mixing bowl. Mix in eggs, then mix in vanilla. Gradually add in dry ingredients until incorporated. Remove dough from the bowl and wrap in plastic wrap. Refrigerate for at least 3 hours.
Preheat the oven to 350 degrees F (175 degrees C). Line 2 cookie sheets with parchment paper.
Remove dough from the refrigerator and roll out to desired thickness on a lightly floured surface. Sprinkle more flour on top of and under the dough, and on the rolling pin as needed to help prevent dough from sticking to the surface. Cut cookies with a round cookie cutter and place 2 inches apart on the prepared cookie sheets.
Bake until cookies are firm to the touch, 10 to 12 minutes. Remove from the oven and transfer to a wire rack to cool, about 30 minutes.
Generously apply frosting to cooled cookies with a butter knife. Sprinkle crushed candy canes over top.