Turns out, it’s not too hard to make fresh pita. And given that the fresh ones put store bought ones to shame, it’s certainly a recipe worth trying. Here, credit for the recipe goes to the New York Times, though the non-baker and Baker tripled the recipe and veered away (mostly by accident) from the initial instructions.
They began by activating the yeast (which they were supposed to do by adding a bit of flour and water.) Having accidentally added too much flour, they decided to add additional water and a bit of honey and maple syrup as sweeteners. Fortunately the yeast still activated and they then let the dough sitting in a bowl for about an hour.
After an hour, the dough had risen, though they had much more of it than the original recipe had called for (due both to their modifications and their tripling of the recipe.)
From there, it was time to roll out the pitas. Each one was rolled flat (with a rolling pin) until they were about 5-8 inches across, and 1/8-/14 inches thick.
Each pita was cooked for a minute and a half on the first side, then flipped and cooked for a minute on the second side. This provided the pitas with enough heat to puff up, but ensured that they did not burn. After initially keeping the still-hot pitas under a clean tea towel, they were transferred to bread baskets.