While chicken marinates, heat your beer to 98℉/36℃ in a small sauce pan. Then, in a food processor, combine flour, yeast, salt, sugar, and olive oil. Pulse about 5 times to mix. With the processor running, slowly stream in the warmed beer until the dough forms. It will be shaggy.
Flip the dough onto a floured surface and knead for about a minute or two, until it becomes slightly elastic.
Cut the dough into 4 equal pieces (about 150g each). Shape each into a ball, cover with a damp paper towel, and let them rest at room temp for 15 minutes.
After 15 minutes, lightly flour a surface and flatten a dough ball with your hands slightly. Then switch to a rolling pin and roll dough into a 6-8 inch round. Repeat this with the next 3 dough balls.
Once the dough is rolled out, allow the pita to rise at room temperature for 30 minutes.