Thick, moist, fluffy, flavorful pitas
I accidentally made a thick pita by simply rolling my pitas thicker.  In my experience, technique is king with pita.  The temperature of the oven (I bake at 450 degrees) and making sure to roll smoothly without any punctures in the dough, then place directly on the rack make perfectly puffy pitas every time.  The thickness of the dough surrounding the pocket is entirely dependent - in my case - on how thick I roll.  I use the lavash cracker recipe from Peter Reinhart's Bread Baker's Apprentice as my dough.