Preheat the oven to 375°F. Line 2 baking sheets with parchment paper.

Put 1-1/3 cups of the semisweet chocolate chips in a microwave-safe bowl. Heat on high for 1 minute, stir the chips, then heat again until melted, 30 to 60 seconds. Stir until smooth. Let cool slightly.

In a stand mixer fitted with a paddle attachment or with a hand mixer, beat the butter. Add the sugar and continue beating until light and fluffy. Add the eggs, vanilla, and 1 teaspoon of the peppermint extract. Beat until smooth, scrape down the sides of the bowl, and beat again. Stir in the cooled, melted chocolate.

In a small bowl, whisk together the flour, cocoa powder, baking powder, xanthan gum, and salt. Add the flour mixture to the butter-sugar mixture and beat until combined. Stir in the remaining chocolate chips. Cover the dough with plastic wrap and refrigerate for 30 minutes.

Scoop the dough with a medium-size cookie scoop or hand-shape walnut-size balls and place them 2 inches apart on the prepared baking sheets. Bake until the cookies are slightly cracked on the top but still slightly yield to the touch, 10 to 12 minutes. Let cool on the pans for a few minutes, then transfer to a cooling rack to cool completely.

Place the white chocolate chips in a microwave-safe bowl. Heat on high in the microwave for 1 minute, stir, and heat again until melted, about 30 seconds. Stir until smooth. Stir in the remaining ¼ teaspoon peppermint extract. Pour the melted white chocolate into a small piping bag or plastic sandwich bag. Snip off one corner and drizzle over the cookies in a zigzag pattern. Sprinkle the crushed candy canes over the cookies before the chocolate sets.