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In a large mixing bowl add 1 cup lukewarm water and stir in yeast and sugar until dissolved. Add 1/2 cup flour and whisk together. Place the mixing bowl in a warm place, uncovered to form a lose sponge. Give it 15 minutes or so, the mixture should bubble.
Now add salt, olive oil and almost all the remaining flour (keep about 1/2 cup of the flour for dusting later). Stir until mixture forms a shaggy mass (at this point, the dough has little to no gluten development and just looks like a sticky mess and you can easily pull bits off). Dust with a little flour, then knead the mixture inside the bowl for about a minute to incorporate any stray bits.
Dust a clean working surface with just a little bit of flour. Knead lightly for a couple minutes or so until smooth. Cover and let the dough rest for 10 minutes, then knead again for a couple more minutes. The dough should be a little bit moist, you can help it with a little dusting of flour, but be careful not to add too much flour.
Clean the mixing bowl and coat it lightly with extra virgin olive oil and put the dough back in the bowl. Turn the dough a couple times in the bowl to coat with the olive oil. Cover the mixing bowl tightly with plastic wrap then lay a kitchen towel over. Put the bowl in a warm place. Leave it alone for 1 hour or until the dough rises to double its size.
Deflate the dough and place it on a clean work surface. Divide the dough into 7 to 8 equal pieces and shape them into balls. Cover with a towel and leave them for 10 minutes or so to rest.
Using a floured rolling pin, roll one of the pieces into a circle that’s 8-9 inches wide and about a quarter inch thick. It helps to lift and turn the dough frequently as you roll so that dough doesn’t stick to your counter too much. (If dough starts to stick, sprinkle a tiny bit of flour). If the dough starts to spring back, set it aside to rest for a few minutes, then continue rolling. Repeat with the other pieces of dough. (Once you get going, you can be cooking one pita while rolling another, if you like). You have two options for baking the pita from here.
Heat the oven to 475 degrees F and place a heavy-duty baking pan or large cast iron skillet on the middle rack to heat. Working in batches, place the rolled-out pitas directly on the hot baking baking sheet (I was only able to fit 2 at a time). Bake for 2 minutes on one side, and then, using a pair of tongs, carefully turn pita over to bake for 1 minute on the other side. The pita will puff nicely and should be ready. Remove from the oven and cover the baked pitas with a clean towel while you work on the rest of the pitas.
To cook pita on stovetop: Heat a cast iron skillet over medium-high heat. (Test by adding a couple drops of water to the skillet, the skillet is ready when the beads of water sizzle immediately). Drizzle a tiny bit of extra virgin olive oil and wipe off any excess. Working with one pita at a time, lay a rolled-out pita on the skillet and bake for 30 seconds, until bubbles start to form. Using a spatula, flip the pita over and cook for 1-2 minutes on the other side, until large toasted spots appear on the underside. Flip again and cook another 1-2 minutes to toast the other side. The pita is ready when it puffs up forming a pocket (sometimes, with this method, the pita may not puff or may only form a small pocket. Try pressing the surface of the pita gently with a clean towel). Keep baked pita covered with a clean towel while you work on the rest.