Combine water, yeast, and a pinch of the sugar in the bowl of a standing electric mixer, stirring until the yeast dissolves. Let sit until bubbly, about 5 to 10 minutes.
Add sugar, whole wheat flour, one cup of the all-purpose or bread flour, salt, yogurt, and oil to the yeast mixture. Mix with paddle attachment until smooth and creamy, about 1 minute. Add 1 1/2 cups more flour, 1/2 cup at a time, until a shaggy dough forms.
Replace the paddle with the dough hook and knead the dough on medium speed until smooth and glossy, about 8 minutes. The dough will be sticky. While kneading, add flour in 1-tablespoon increments if necessary for the dough to clear the sides of the bowl (the dough will stick to the bottom of the bowl). Scrape down the sides of the bowl with a rubber spatula as necessary.
Transfer the dough into an oiled bowl, flipping it to coat the entire surface with oil. Cover with plastic wrap and let the dough rise until doubled in size, about one hour. (At this point the dough can be refrigerated for up to 2 days).
Turn the dough onto a lightly floured surface and sprinkle it with flour. Divide the dough into 8 equal portions. Shape each portion into a round ball. Roll each ball into a 4-inch circle, let rest for 10 minutes, then roll each disk into a 6-inch circle.
About 5 minutes before cooking the flatbreads, heat a large, heavy skillet over medium-high heat. Working one at a time, lift the dough circles, gently stretching each one about 1 inch larger, before placing it on the skillet. Cook until bubbles appear on the surface of the dough, about 30 seconds. With tongs, flip the bread and cook until the bottom is speckled and deep golden brown in spots, 1 to 2 minutes longer.
Transfer to wire rack and let cool about 5 minutes. Wrap the breads loosely in a clean kitchen towel and serve warm, or allow them to cool completely to use for sandwiches.