Combine the barley flour, honey, instant yeast, and water in a mixing bowl. Let this sponge sit for twenty minutes until small bubbles begin to appear on the surface.
Add the all-purpose flour, salt, and olive oil. Stir together just until a shaggy dough begins to form. Let rest for ten minutes.
After resting, turn dough onto the counter and knead for ten minutes. Using the heel of your hand, push firmly down and forward, rotate 90°, fold in half, and repeat. Continue this motion for ten minutes until the dough becomes smooth.
Once your dough has come together in a smooth ball, place in a well-oiled bowl, cover loosely with a dishcloth or plastic wrap, and let rest for an hour.
Divide your dough into ten even pieces. Shape each piece into a ball by cupping your hands and rotating between your palms. Once shaped, let the rounds rest another hour while preheating the oven to 400°F.
Place an empty baking sheet in the oven. Using a rolling pin, roll the dough one piece at a time into 6-inch circles, rotating 90° after each roll. Let rest for 5 minutes.
Quickly place rounds onto the hot baking sheet and return to the oven. Bake for 4 minutes. Be careful when you remove your pita from the oven – while baking, the bread will puff with steam, which can severely burn you if deflated while still hot.
Serve pita with traditional Greek mezze, such as hummus or tzatziki, as a wrap for a sandwich, or the crust of a pizza. Or eat it on its own, savoring the complex balance of flavors.