In the bowl of a standing mixer fitted with a dough hook, dissolve the yeast in the lukewarm water. Let stand, stirring, until foamy, about 5 minutes. Add the all-purpose flour and mix on low speed for 1 minute. Cover the bowl with a kitchen towel and let stand for 1 hour.
Add the oil and salt to the sponge and gradually beat in the whole wheat flour on low speed. Increase the speed to high and knead until the dough is smooth, about 3 minutes. Cover and let stand until the dough has doubled in bulk, 1 1/2 to 2 hours.
Divide the dough into 16 pieces and cover with a dish towel. On a lightly floured work surface, roll out each piece to a 7-inch round, about 1\8 inch thick. Stack the rounds between sheets of plastic wrap and cover with a dish towel.
Heat 2 lightly oiled cast-iron skillets. Place 1 dough round in each skillet and cook over moderately high heat for 20 seconds. Turn and cook until big bubbles appear in the pitas, about 1 minute. Turn again and cook until the pitas puff, 1 to 2 minutes longer. Wrap the pitas in a towel and keep warm while you cook the rest. Lightly oil the skillet as needed. Serve the pitas warm.