Place the candy canes or peppermint candies in a zip-top bag and tap with a rolling pin until you have a mix of fine dust and small (pea-size) bits; set aside 2 tablespoons for finishing.
In a medium bowl, whisk together the flour, cocoa powder, baking soda, salt, and espresso powder until evenly combined.
Preheat the oven to 325°F. Line two baking sheets with parchment paper.
In a large bowl, cream the softened butter with the brown sugar and granulated sugar on medium-high speed until pale and fluffy, 2 to 3 minutes, scraping the bowl once halfway through.
Beat in the egg, vanilla, and peppermint extract just until smooth, about 30 seconds. Do not overmix after the egg goes in.
Add the dry ingredients in two additions on low speed, mixing only until no dry streaks remain. Fold in the white chocolate chips and all but the reserved 2 tablespoons of crushed peppermint.
Cover and chill the dough for 20 minutes to help the cookies bake thicker and keep the peppermint pieces from melting too quickly.
Scoop the dough into 1½-tablespoon portions (a #40 scoop works well) and set them 2 inches apart on the prepared sheets.
Bake one sheet at a time for 12 to 14 minutes, rotating the pan once halfway through. The edges should look set while the centers remain soft and slightly puffed.
Immediately after removing from the oven, gently press a few extra white chocolate chips into the tops and sprinkle with the reserved peppermint so it adheres to the warm cookies.
Cool on the baking sheet for 5 minutes, then transfer to a rack to finish cooling. Repeat with remaining dough.