Add the Caputo Fioreglut, brown rice flour, sorghum flour, salt, and yeast to the bowl of a stand mixer fitted with the paddle attachment, whisk to combine.
With the mixer on low speed, slowly pour in the warmed water and olive oil. Increase to medium speed and mix for 2 minutes, scraping the bowl as needed. The batter will be thick and sticky.
Scoop the dough into a ball and coat it with a light layer of olive oil. Cover and let rise in a warm spot for 45 minutes, or until doubled in size.
Oven: Preheat a pizza stone, baking steel, or cast iron skillet in the oven at 250°C for at least 30 minutes.
While the oven preheats, shape the pitas. Turn the dough out onto a surface lightly dusted with brown rice flour. Divide into 4 equal pieces (about 120g each). Gently knead each piece and form into a smooth ball.
Use a rolling pin to roll each piece into a 6-inch round, about ¼-inch thick. Roll each one only once.
Cover the shaped pitas with a kitchen towel or plastic wrap and let rest at room temperature for 15–20 minutes.
Oven: Place one pita at a time directly on the hot stone. Bake for 5 minutes total. After it has fully puffed, flip it once halfway through to brown the other side, if desired.