Preheat oven to 350°F. Line two baking sheets with parchment.
Cream butter and sugar until pale and fluffy, about 2–3 minutes.
Beat in the egg, vanilla, and peppermint until smooth.
Whisk flour, baking powder, baking soda, and salt in a separate bowl.
Add dry ingredients to the butter mixture in two additions. Mix on low just combined.
Fold in about half of the crushed candy canes.
Scoop 1½-tablespoon portions and roll into balls. Roll in sanding sugar if using.
Arrange 2 inches apart on sheets. Gently flatten each ball slightly.
Press remaining candy cane bits on top of each cookie.
Bake 8–10 minutes until edges set and centers look pale.
Cool 5 minutes on the sheet, then move to a rack to finish cooling.