Start with warm water in a large mixing bowl. Add yeast and sugar. Stir to dissolve. Add the 1/4 cup all-purpose flour and whisk together. Put the bowl in a warm (not hot) place, uncovered, until mixture is frothy and bubbling, about 15 minutes.
Add salt, olive oil, and nearly all remaining all-purpose flour (reserve 1/2 cup). With a wooden spoon or a pair of chopsticks, stir until mixture forms a shaggy mass. Dust with a little reserved flour, then knead in bowl for 1 minute, incorporating any stray bits of dry dough.
Turn dough onto work surface. Knead lightly for 2 minutes, until smooth. Cover and let it rest for 10 minutes, then knead again for 2 minutes. Try not to add too much reserved flour; the dough should be soft and a bit moist. (At this point, the dough may be refrigerated in a large zippered plastic bag for several hours or overnight. Bring dough back to room temperature, knead into a ball and proceed with the recipe.)
Clean the mixing bowl and put the dough back in it. Cover bowl tightly with plastic wrap, then cover with a towel. Put the bowl in a warm (not hot) place. Leave until dough has doubled in size, about 1 hour.
Heat oven to 475 degrees F. On the bottom shelf of the oven, place a heavy-duty baking sheet, large cast-iron pan or ceramic baking tile. Punch down dough and divide into 8 pieces of equal size. Form each piece into a little ball. Place dough balls on the work surface, cover with a damp towel and leave for 10 minutes.
Remove 1 ball (keeping others covered) and press into a flat disk with a rolling pin. Roll to a 6-inch circle, then to an 8-inch diameter, about 1/8 inch thick, dusting with flour if necessary. (The dough will shrink a bit while baking.)
Carefully lift the dough circle and place quickly on a hot baking sheet. After 2 minutes the dough should be nicely puffed. Turn over with tongs or spatula and bake for 1 more minute. The pita should be pale, with only a few brown speckles. Transfer warm pita to a napkin-lined basket and cover so bread stays soft. Repeat with the rest of the dough ball.