Place racks in upper and lower thirds of oven and preheat to 350°F.

Seal candy canes in bag and coarsely crush with a rolling pin or a skillet. Set aside.

Heat butter in a small saucepan over medium, stirring constantly, until it foams, then browns, about 4 minutes.

Transfer browned butter to a large heatproof bowl.

Add brown sugar, granulated sugar, and 1 Tbsp. water, and whisk vigorously until sugar is dissolved and mixture is smooth and elastic, about 2 minutes.

Add egg, vanilla extract, and peppermint extract and whisk until smooth, about 1 minute.

Add flour, cocoa powder, salt, baking powder, and baking soda and mix with a rubber spatula until soft dough forms and no pockets of dry flour remain.

Let dough sit, uncovered, at room temperature 15 minutes. (This will allow it to stiffen slightly, making rolling it into balls easier.)

Scoop dough by rounded 2-Tbsp. portions (#30 cookie scoop) and roll into smooth balls, onto 2 parchment-lined baking sheets, spacing at least 2 inches apart (about 7 per sheet).

Bake cookies, rotating pans top to bottom and front to back halfway through, until puffed slightly and cracked have formed, 8-10 minutes.

While cookies are baking, cut or break white chocolate into irregular angular shards that will fit nicely in the centers of cookies (about 1” x 1”).

Let cookies cool on baking sheets 2 minutes. (They should be very warm but not piping hot).

gently press chocolate pieces into centers of cookies (they should sit almost flush with tops of cookies; you can use two pieces in a cookie if shards are small), then return to oven and bake just until tops of chocolate pieces are barely melted, about 30 seconds.

Immediately sprinkle reserved candy canes on top. Let cookies sit on baking sheets at room temperature four hours to allow chocolate to harden. (Or, chill 30 minutes to speed up the process.)

DO AHEAD: Cookies can be baked four days ahead. Store airtight at room temperature.