In a large bowl, combine sieved whole wheat flour, 3 cups all-purpose flour and salt. Using a wooden spoon or the dough hook of a stand mixer, add the liquid ingredients and mix thoroughly, only adding the additional 1/2 cup flour if the dough seems especially sticky.
Turn out onto floured surface and knead for about 5 minutes, using a scraper if necessary to lift and pull the dough over itself until it feels smooth and elastic, about 5 minutes. Try to resist adding additional flour, although humid conditions may require a bit more. This is a soft, tender dough.
Lightly coat a medium bowl with oil and place the dough face down, then flip so the coated side is on top. Cover with plastic wrap and let rise in a warm place until doubled in size, about 45 minutes.
When dough is ready, begin heating griddle on stovetop. If using the oven, preheat to 500 degrees. If you have a pizza stone, place that on the bottom rack to heat. Otherwise, place a heavy sheet pan in the oven. (You want to lay the flatbread dough on a hot surface to help it puff.)
Turn the risen dough out onto a lightly floured surface and divide into 8 pieces, forming each into balls. Cover with a cloth and let rest for 5 minutes.
With a rolling pin on a floured surface, roll a ball into a flat circle about 8 inches across, flipping once. If the dough shrinks and pulls back, let rest for a minute, then continue to roll. Brush lightly with water and sprinkle with a scant teaspoon of sesame seeds. Roll once with the pin to help seeds adhere.