Whisk together the gluten-free flour mixture, baking powder, and salt in a medium bowl. Set aside.
Cream the sugar and butter together in a stand mixer fitted with the paddle attachment on medium-low speed, scraping down the sides of the bowl as needed, until light and fluffy, about 3 minutes. Add the eggs and extracts and continue mixing on medium-low until just combined, about 15 seconds more. Reduce the speed to low, add the flour mixture and continue mixing until it is completely incorporated, about 30 seconds.
Divide the dough in half, shape into 4-by-6-inch rectangles, and wrap in plastic. Refrigerate for at least 6 hours or up to overnight.
When you’re ready to bake, remove half of the dough from the fridge and let sit on the counter for 10 minutes to soften. (If refrigerated overnight, it may take longer than 10 minutes to soften enough to roll.) Line two baking sheets with parchment paper.
Lightly dust a third piece of parchment paper with powdered sugar, place the doughDon't dust the top or the resulting cookies won't be smooth. in the middle, then top with a fourth sheet of parchment. Roll the dough into a rough rectangle that’s 1/4-inch thick. Punch the dough into shapes using cookie cutters.
One at a time, use a small offset spatula to transfer the cookies to the prepared sheet pans. (I use a small metal offset spatula for this). If the dough becomes too soft to handle during any point of the process, chill in the fridge for 10 minutes to firm. Repeat with the second rectangle of dough and second baking sheet. Re-roll the dough as needed to make as many cookies as you’d like. Refrigerate for 30 minutes to firm up.
Meanwhile, place one oven rack in the top third of the oven and another in the bottom third and crank the oven to 325°F.
Bake the cookies straight from the fridge for 8 minutes, then rotate the pans 180 degrees and swap the top with the bottom pan. Continue baking until the edges of the cookies just begin to brown, about 7 more minutes. Move the sheet pans to wire cooling racks and let cool 10 minutes. Slide the parchment off the pan onto the rack and let the cookies cool completely before icing. Repeat the rolling, cutting, chilling, and baking steps with the remaining dough.