Make sponge: In a large mixing bowl add the 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.
Form the pita dough: 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.
Knead the dough: 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.
Let the dough rise. 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.
Divide the dough. Deflate the dough and place it on a clean work surface. Divide the dough into 7 to 8 equal pieces (~85g) and shape them into balls. Cover with a towel and leave them for 10 minutes or so to rest.
Shape the pitas. 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.
Cook in pizza oven