Preheat your oven to 350°F and line a baking sheet with parchment paper. Set aside for later.
In a medium bowl, cream the butter and brown sugar together until the mixture becomes pale and fluffy, about 2 minutes. Beat in the egg, vanilla extract, and peppermint extract until fully combined.
In a separate bowl, whisk together the flour, cocoa powder, baking soda, and salt. Fold the dry ingredients into the wet ingredients just until a dough forms.
Cut each piece of peppermint bark into quarters so they are easier to tuck into the dough.
Scoop about 2 tablespoons of dough, flatten slightly, and place 2 to 3 pieces of peppermint bark in the center. Roll the dough around the filling to seal completely. Place the dough balls on the baking sheet, leaving about 2 inches of space between each cookie.
Bake for 10 to 12 minutes, until the edges are set but the centers are still soft. Let the cookies cool on the baking sheet for 5 minutes, then transfer to a cooling rack.
While the cookies cool, melt the white chocolate in a microwave-safe bowl, heating in 30-second intervals and stirring between each round, until smooth.
Dip the tops of the cooled cookies into the melted chocolate and sprinkle with crushed peppermint. Let the chocolate set before serving.