Place one oven rack in the top third of the oven and another in the bottom third and crank oven to 325 degrees F.
Stir together the flour mix, baking powder and salt.
Install the paddle on your stand mixer and cream the sugar and butter together on medium-low speed for about 3 minutes, scraping down the side of the bowl as needed, until the sugar is no longer visible. Add the eggs and extracts and continue mixing on medium-low until just combined, about 15 seconds more. Reduce speed to low, add the dry mixture to the bowl and continue mixing until its completely incorporated, about 30 seconds, but don't overmix.
Divide the dough in two, shape into 4-by-6-inch rectangles and wrap in plastic. Refrigerate overnight or at least 6 hours.
Remove dough from fridge 10 minutes before rolling to soften. Lightly dust a 12-by-16-inch piece of parchment with powdered sugar, place the dough in the middle, then top with a second sheet. (Don't dust the top or the resulting cookies won't be smooth.) Roll dough to one quarter-inch thickness, then punch dough into shapes with cookie cutters.
Transfer cookies one at a time to a parchment-lined sheet pan (I use a small metal offset spatula for this) then refrigerate for 30 minutes before baking. 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.
Bake the cookies for 8 minutes then rotate the pans swapping the top pan with the bottom pan and rotating both 180 degrees. Continue baking until the edges of cookies just begin to brown, about 7 minutes. Move the sheet pan to a rack and cool 10 minutes, then slide the parchment off the pan onto the rack and cool completely before icing.
Combine all the ingredients in a large, airtight container and use within 6 months.
Fit the paddle attachment onto your stand mixer and combine the egg whites, cream of tartar and salt on low speed, then slowly introduce the sugar. Once it's all worked in, increase speed to medium high and beat until frosting-like, about 5 minutes. If a thinner consistency is desired, add a small amount of water (or rum). Use immediately or store in an airtight container and refrigerate.
*Work granulated sugar in a blender at high speed until it's superfine, about 2 minutes.