Preheat the oven. Preheat to 350°F (177°C). Line 2 baking sheets with parchment paper.
Mix the dry ingredients. In a large bowl, whisk together 2 cups (280g) gluten-free flour blend, 2/3 cup (65g) cocoa powder, 1 teaspoon baking soda, and 1/2 teaspoon salt. Set aside.
Cream butter and sugars. In another large bowl, cream 1 cup (226g) butter, 1 cup (200g) granulated sugar, and 3/4 cup (150g) brown sugar with an electric mixer until light and fluffy, about 2-3 minutes.
Add eggs and extracts. Mix in 2 eggs, 1 teaspoon vanilla extract, and 1 teaspoon peppermint extract until combined.
Add dry ingredients. Add the dry ingredients to the wet ingredients a little at a time and mix until just combined.
Fold in chocolate chunks. Stir in 1 1/2 cups of the chocolate chunks, reserving 1/2 cup for topping.
Scoop the dough. Scoop 2.5-tablespoon portions of dough onto the baking sheets, leaving at least 2 inches between each.
Top with extra chocolate. Press the remaining chocolate chunks onto the tops of the dough balls.
Bake. Bake for 10-12 minutes, until the edges are set and the centers still look slightly soft. Cool on the pan for 5 minutes, then transfer to a wire rack to cool completely.
Dip in white chocolate. Melt 8 ounces white chocolate in 20-30 second intervals in the microwave, stirring between each. If it's not melting smoothly, add 1 teaspoon vegetable oil. Dip half of each cooled cookie and place on parchment.
Add crushed peppermint. While the white chocolate is still wet, sprinkle with crushed candy canes. Let set completely.