Put 1 cup of lukewarm water in a large mixing bowl. Add yeast and sugar. Stir to dissolve. Add the whole-wheat flour and 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 rest 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 recipe.)

Clean the mixing bowl and put the dough back in it. Cover the 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 a cast-iron skillet on medium-high heat. 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 in a hot skillet. After about 2 minutes the dough should be nicely puffed. Turn over with tongs or spatula. The pita should be pale, with some brown speckles. Transfer warm pita to a napkin-lined basket and cover so bread stays soft. Repeat with the rest of the dough balls.