Use a mixer with beaters or the paddle attachment to beat the butter and both sugars together until light and fluffy. If you don't have an electric mixer, use a wooden spoon or your hands to ensure the sugars get dissolved in the butter.
Beat in the extracts and egg, switching to a whisk if you are doing this by hand.
Place the gluten-free flour, baking powder, baking soda, and salt in the bowl. Lightly stir them together on top, then gently mix together until there are no dry spots. The dough will be fairly thick. Save 16 large pieces of white chocolate for topping the cookies. Fold in the remaining chopped white chocolate and crushed candy canes until thoroughly combined.
Divide the dough into 14 balls, each about 2-3 tablespoon of dough (approximately 60g). Place on a small parchment-lined baking sheet or plate and top each ball with a large piece of white chocolate. Refrigerate for 30 minutes. Halfway through the chill time, preheat the oven to 350ºF.
Place the dough balls on 2 large parchment-lined baking sheets. Make sure there is room between them for the cookies to spread. Bake for 13-15 minutes until the edges are lightly coloured and the centres are still a little soft. For perfectly round cookies, take a large cookie cutter or the top of a glass and gently run it around each cookie immediately after removing it from the oven. This will reinforce a circular shape as the cookie cools. Sprinkle the tops with more crushed candy cane pieces.
Allow the cookies to cool on the baking sheet for 15 minutes until they have set slightly, then transfer to a wire rack to continue cooling. Enjoy warm or allow to come to room temperature.