Place a pizza stone under the broiler; heat for 30 minutes.
Working in 4 batches, dust 1 ball dough with semolina.
Using your fingertips, press dough into a 10" circle about 1⁄4" thick, leaving a 1" crust around the edges.
Hold dough straight up, and with fingertips circling crust, slide fingers around crust in a circular motion as you would turn a steering wheel until dough in the center is stretched to about 1⁄8" thick; transfer to a semolina-dusted pizza peel.
Spread 1⁄2 cup sauce over dough, and distribute a quarter each of the cheese and basil leaves; drizzle with oil.
Slide pizza onto stone; broil until cheese melts and crust is puffed and charred in spots, 3-4 minutes. Serve hot.