Whisk together the gluten-free flour mixture, baking, powder, and salt in a medium bowl. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, cream together butter and sugar on medium-low speed until lighter in color and fluffy, about 3 minutes, scraping down the sides of the bowl with a silicone spatula. Add egg and extracts and continue to mix on medium-low until just combined, about 15 seconds more. Add the dry mixture to the bowl and continue mixing until it’s completely incorporated, about 30 seconds, but don’t over mix.
Divide the dough in two, wrap in plastic and shape into two 4 1/2-by-6-inch rectangles. Refrigerate at least 6 hours, overnight is better.
When you're ready to bake, place one rack in the top third of the oven and another in the bottom third and heat the oven to 325°F. Line two baking sheets with parchment paper.
Remove dough from fridge and let sit for 10 minutes to soften enough to roll. Between two sheets of parchment paper, roll dough out to 1/4-inch thickness; lightly dust the bottom sheet with powdered sugar, but don’t use any sugar on top of the dough, or the cookies will have an uneven surface once baked.
Punch dough into shapes with metal cookie cutters. Using a small, offset spatula, transfer cookies one at a time to a parchment-lined sheet pan and then refrigerate for another 30 minutes before baking. If the dough becomes too soft to handle during rolling, pop it in the fridge for 10 minutes to firm up before continuing to roll and cut. Repeat with the second rectangle of dough.
Bake the cookies for 8 minutes before rotating pans—swap the top pan with the bottom pan and rotate both pans 180 degrees—then bake until the edges of the cookies are just beginning to brown, about 7 minutes more. Remove the sheet pans to a rack and let cool 10 minutes before taking the cookies off the sheet and letting them cool completely on the rack before icing.