Make the sponge: combine 1 cup lukewarm water, yeast, and sugar in a large bowl. Stir to dissolve. Add whole-wheat flour and ¼ cup all-purpose flour; whisk. Leave in a warm place uncovered for 15 minutes until frothy and bubbly.
Add salt, olive oil, and nearly all the remaining flour (reserve ½ cup). Stir until a shaggy mass forms. Knead in the bowl 1 minute, then turn out and knead on a surface 2 minutes until smooth. Rest 10 minutes, then knead 2 more minutes. Dough should be soft and slightly moist.
Return to a cleaned bowl, cover tightly with plastic wrap and a towel. Leave in a warm spot for 1 hour until doubled.
Heat oven to 475°F with a heavy baking sheet, cast-iron pan, or ceramic tile on the bottom shelf.
Punch down dough and divide into 8 balls. Cover and rest 10 minutes. Roll each ball to an 8-inch circle, about ⅛ inch thick.
Carefully place a dough circle on the hot baking surface. Bake 2 minutes until puffed, then flip and bake 1 minute more. Pita should be pale with a few brown speckles. Transfer to a napkin-lined basket and cover. Repeat.
