Heat your grill on high.
Flatten a dough ball into a 6” X 8” oval. Pour 2 tablespoons of olive oil on each baking sheet and place a dough oval on it, continuing to stretch the dough to the thickness you prefer. (Alternatively, you may simply roll out the dough as in the method above.)
Use tongs to dip the rag in oil and generously slather it over hot grill grates to oil well.
Place pizza bases onto the grill. Immediately turn the heat down to medium and close lid, cooking until bottoms are browning but not burned, about 2 minutes.
Brush the tops of the pizzas with oil and flip with tongs. Grill the second side. Note: if pizza base sticks, wait a bit. As water evaporates from the dough it will release.
Remove pizza bases from grill back onto baking sheets. Adjust grill burners so that half is on low and half on medium heat.
Cover pizza bases with prepared topping, dotting with mozzarella.
Finish pizzas by sliding onto low side of grill and baking, lid closed, 1 to 2 minutes, or until toppings are hot and cheese is melted.
Remove to a cutting board, slice, and serve.
Repeat with remaining dough.