Butter 2 8-inch square pans or 2 8-inch round cake pans and line them with parchment paper.
In a large bowl, beat together the butter, maple sugar, and salt until well combined
Mix in the flour to form a smooth dough. Don't worry if the dough is crumbly at first - it'll come together as it's mixed.
Divide the dough in half and press each half into the prepared pans (alternatively, wrap one half of the dough in plastic and set it aside to bake when the first round is done or refrigerate it for later). Prick the dough all over with a fork so that any steam that forms is released during baking.
Preheat the oven to 325℉, then bake the shortbread for 40-45 minutes, until the surface is set and slightly golden brown (the edges will be a darker golden brown). Remove from the oven and immediately turn it out onto a clean work surface. Gently brush the shortbread all over with the maple syrup.
Using a sharp knife or a pizza cutter, divide the shortbread into pieces. (I cut 16 pieces from each pan.) Then transfer the pieces to cool completely on a rack.
Notes
Rest the dough. Let the dough rest in the fridge while the oven heats so the gluten-free flour hydrates properly.
Make sure to chill the dough. Place the dough in the pans in the fridge for up to 30 minutes to let it chill. This will help the butter to resolidify, creating gorgeously flaky cookies later.
Bake half now, and bake half another time. You can bake off half of the shortbread dough and save the other half for later. Divide the dough and form a ball with the dough you're not ready to use. Wrap it well in plastic wrap and store it in the fridge for up to 3 days, or freeze it for up to 3 months. If you freeze the dough, thaw it in the fridge before placing it in the pan.
You don't have to use maple sugar. If you can't find maple sugar, no worries! You can use granulated or light brown sugar. But don't skip the maple syrup over the top of the cookies - that's where you'll get that lovely maple flavor.
Sprinkle with flaky salt. If you're a salt-lover like me, you may want to sprinkle your finished shortbread with flaky salt for a sweet-salty combo.
Another flour. Try using another gluten free flour such as oat flour or sorghum flour for the gluten-free all-purpose flour blend (use 300 grams of whatever flour you choose).
Make them dairy-free. Swap the butter for non-dairy butter.
Add nuts. To add a little crunch to maple shortbread cookies, add 1 cup of finely chopped nuts like almonds, walnuts, or pecans to the dough.