Stir together warm water and sugar in a measuring cup, then sprinkle yeast over mixture and let stand until foamy, about 10 minutes (If yeast doesn't foam, discard and start over with new yeast.
Stir together flour (3 3/4 cups) salt, and baking powder in a large bowl. Stir together yogurt, egg, and oil in a small bowl. Make a well in center of flour mixture and add yeast mixture and yogurt mixture, then stir in flour mixture gradually with a wooden spoon until dough just comes together.
Knead dough on a lightly floured surface with floured hands, lightly reflouring work surface and your hands when dough becomes too sticky, using no more than 1/4 cups flour, until dough is smoother, soft, and elastic, about 10 minutes. Transfer dough to a lightly oiled large bowl and cover bowl with plastic wrap. Let dough rise in a draft-free place at warm room temperature until doubled in bulk, 1 1/2 to 2 1/2 hours.
Make Topping While Dough Rises: Put oven rack with pizza stone in lower third of oven and preheat oven to 450 degrees (heating pizza stone will take about 1 hour).
Coarsely grate onion on largest holes of a box grater and drain well in a sieve. Cook onion and sesame seeds in oil in a 10-inch heavy skillet over moderate heat, stirring occasionally, until onion is just beginning to turn golden, about 12 minutes. Transfer to a bowl and cool to room temperature.
Turn dough out onto a lightly floured surface without punching down. Cut dough into 2 pieces and trasfer 1 piece to parchment lined pizza peel or baking sheet. (Keep remaining piece of dough covered with plastic wrap).
Flatten dough with lightly oiled fingertips, patting and gently stretching into an 11 to 12 inch round.
Spread half of onion mixture over round, then poke holes through dough all over with skewer.
Slide bread, along with parchment, onto pizza stone. Bake until golden brown, 10 to 12 minutes. Remove bread to a rack and discard parchment. While first flatbread is baking, make another bread on another sheet of parchment, then bake in same manner. Serve flatbreads warm.