Place a baking stone on middle shelf of oven; preheat oven to highest setting for at least 1 hour.
*Make pizza one at a time: shape dough on floured surface; transfer it to pizza peel or inverted sheet pan that has been dusted with flour.
Spread half of the mozzarella over the surface of the dough; distribute half the tomatoes over the cheese; sprinkle half the grated Parmigiano-Reggiano over the top.
Carefully slide the pizza from the peel onto the baking stone; bake for 7-9 minutes.
When pizza is done, the crust should be puffy and slightly charred on the edge and thinner in the center, and the cheese should be golden; the underside of the crust should be brown and crisp, not white and soft.
While the pizza is baking, stack the arugula leaves, roll them up lengthwise and cut thinly crosswise to create shreds.
Remove the finished pizza from oven and immediately sprinkle half of the arugula over the top; let pizza cool for about 3 minutes so that the arugula can begin to wilt, then serve it whole or in slices.
Repeat with the remaining ingredients to make the second pizza.