Beat the butter and powdered sugar using a mixer for 5-7 minutes. Add granulated sugar and sift in the gluten-free all-purpose flour. Mix for 2 minutes. Add egg, ground hazelnuts, cinnamon, and vanilla extract. Knead into a ball with your hands. Wrap in plastic wrap and refrigerate for 1 hour.
Spread the dough onto a floured surface and roll out to about ⅓ inch thick. It’s best to work in batches and keep the rest of the dough in the fridge. Cut out as many cookies as will fit with a round cookie cutter.
Transfer to a baking sheet lined with parchment paper and bake for 15 minutes. Cool before decorating. Using a toothpick dipped in melted chocolate, make a deer horn-like decoration on each cookie. Then, make a drop of chocolate on each cookie where the nose should be and immediately glue a red nose from sugar pearl sprinkles or cinnamon imperial candy in its place. Set aside to firm before serving.