Preheat the oven to 350 degrees F (175 degrees C). Line 2 cookie sheets with parchment paper.

Sift together flour, cocoa powder, baking soda, and salt into a bowl.

Cream brown sugar and margarine in a bowl with an electric mixer until well blended. Beat in eggs, vanilla, and peppermint extract until light and creamy. Add dry ingredients and mix until just blended. Stir in chocolate chips.

Drop 1/4-cup scoops of dough onto the prepared sheets; cookies should be about 3 inches apart.

Bake in the preheated oven until the edges are lightly toasted, 15 to 17 minutes. Cool on the cookie sheets briefly before transferring to a wire rack to cool completely, about 20 minutes.

Spread melted white chocolate to cover the top of each cookie. Sprinkle crushed candy canes over top. Let white chocolate cool completely before serving, 15 to 20 minutes.