Quarter the dough and shape into balls. Place on a lightly floured surface, cover with a kitchen towel and let stand at room temperature for 30 minutes.
Heat a medium (8-inch) cast-iron skillet over medium for 10 minutes. (Make sure you have a lid.)
Flatten 1 piece of dough, then stretch into an 8-inch round. Add 1 tbsp. EVOO to the skillet, tilting to coat. Carefully lay the dough in the bottom of the pan and cook for 1 1/2 minutes. using a metal spatula, flip the dough, then cover and cook until the bottom is brown, about 3 minutes. Turn the dough again and, working quickly, spread 1/4 cup of the tomato sauce on top. Sprinkle with 1/2 cup of the cheese, top with some of the onions, then cover and cook until the cheese melts, about 1 minute longer.
Transfer the pizza to a cutting board and top with some arugula. Wipe out the pan and repeat with the remaining ingredients.