To prepare dough, weigh or lightly spoon flour into a dry measuring cup and spoons, and level with a knife. Combine 2 tablespoons flour, 3 tablespoons warm water, and yeast in a large bowl; let stand 20 minutes. Add remaining 5 ounces flour (about 1 cup), remaining 1/4 cup warm water, olives, oil, and 1/4 teaspoon salt; stir until smooth. Turn dough out onto a lightly floured surface. Knead until smooth and elastic (about 8 minutes); add enough flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel sticky).
Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 2 hours or until doubled in size. (Gently press two fingers into dough. If indentation remains, dough has risen enough.) Punch dough down; cover and let rest 5 minutes. Divide in half. Working with one portion at a time (cover remaining dough to prevent drying), roll each portion into an 8-inch circle on a floured surface. Place dough on a baking sheet coated with cooking spray and sprinkled with cornmeal.
Position an oven rack in the lowest setting. Place a pizza stone on the lowest rack. Preheat oven to 500°. Preheat pizza stone for 30 minutes before baking dough.
Slide rolled dough onto preheated pizza stone, using a spatula as a guide. Bake on lowest oven rack at 500° for 7 minutes. Remove flatbread to cutting board.
To prepare salad, combine arugula and remaining ingredients in a bowl; toss to coat. Place 1 1/2 cups salad on each flatbread. Fold flatbread over salad; cut piadines in half.
Wine note: The Mediterranean ingredients of this dish call for a wine of the region. Look for whites made from southern Italy's fiano grape, like Mandrarossa Fiano 2008 ($9) from Sicily. —Jeffery Lindenmuth