In a large bowl if mixing by hand or in the bowl of a food processor fitted with a dough hook, dissolve the yeast and sugar in the water. Once it becomes a bit frothy on top, stir in the olive oil, salt, and flour. Mix the dough until smooth, and elastic; about 5 minutes by machine, 10 by hand. Shape the dough into a ball and place in a large lightly oiled bowl. Turn the ball to coat it with oil. Cover with a clean kitchen towel and let stand in a draft-free spot about 1 hour, or until doubled in size.
In a medium bowl, thoroughly combine the topping ingredients, except the feta cheese, together by “smooshing” them with your hands (use a latex glove if you like). This may be made a day in advance of using (cover and refrigerate—remove from refrigerator 30 minutes before using).
Heat the oven to 500 degrees F. (If using a baking stone, place it on the center rack.)
Once the dough has doubled in size, turn it out onto a lightly floured work surface. Gently press the air out of the dough, gently knead it for one minute, and shape it into a 12-inch log. Cut the log into 8 equal pieces, roll those pieces into individual balls. Cover with a clean kitchen towel and let rest 10 to 15 minutes.
Roll each piece out into a very thin, 1/8-inch thick, 8-inch circle or a 12- by 4-inch football shape. If using pizza peel and baking stone, generously sprinkle the peel with cornmeal. Place one dough round onto the peel. Spread about ¼ cup of the filling over the top of the dough, leaving a ¼ inch rim around the outside. Scatter a few teaspoons of crumbled feta on top. Slide the dough onto the center of the baking stone and bake 10 to 15 minutes, or until the bottom is lightly crisped and the topping is fully cooked.
If using baking sheets, line 2 rimmed baking sheets with parchment paper. Depending on the size of the baking sheets, place 2 dough rounds/ovals onto each of the baking sheets. Spread a thin layer, about ¼ cup, of the filling over the top of the dough, leaving a ¼-inch rim around the outside. Bake 10 to 15 minutes, or until the bottom is lightly crisped and the topping is fully cooked.
In a large bowl, combine the salad ingredients, and place on top of the hot flatbreads. Serve immediately.