In a medium bowl, whisk together the gluten-free flour, sorghum flour, xanthan gum, salt, and baking soda until combined. Set aside.
Add the butter, shortening, granulated sugar, and brown sugar to the bowl of a stand mixer. Mix on medium-low speed until combined. Scrape down the bowl and add the egg and vanilla extract and mix well. Add the sourdough discard and mix until completely incorporated.
Pour in the flour mixture and mix on low speed until the dry ingredients are combined. Turn off the mixer and add the chocolate chips, stirring them in by hand.
Form the dough into one large ball in the bottom of the bowl, then cover the bowl with plastic wrap and chill for at least 6 hours (or up to 72 hours).
After the dough has chilled, preheat the oven to 350°F and line 2 baking sheets with parchment paper or silicone baking mats.
Scoop the dough out and form into 24 even-sized balls of cookie dough.
Bake the cookies in batches of 6-8 per sheet on the center rack for 8-12 minutes. Once done, cool the cookies on the baking sheet for 10 minutes before removing and cooling on a wire rack.
¼ cup gluten-free flour can be used in place of the sorghum flour.
Omit the xanthan gum if your flour blend already contains it.
Keep the cookie dough chilled between baking each batch.
Cookies can be frozen once baked. You can also freeze the cookie dough after scooping. Frozen cookie dough can be baked from frozen and will take 10-14 minutes to bake.