Make sponge: Put 1 cup 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 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, dough may 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 dough back in it. Cover bowl tightly with plastic wrap, then cover with a towel. Put bowl in a warm (not hot) place. Leave until dough has doubled in size, about 1 hour.
Heat oven to 475 degrees. On bottom shelf of 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 work surface, cover with a damp towel and leave for 10 minutes.
Remove 1 ball (keeping others covered) and press into a flat diskc with 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 hot baking sheet. After 2 minutes the dough should be nicely puffed. Turn over with tongs or spatula and bake 1 minute more. 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 balls.
Some measurements for dry ingredients are given by weight for greater accuracy. The equivalent measurements by volume are approximate.