1. Place tomatoes, cut side down, in a oiled grill pan.
2. Grill over direct heat until tomatoes are slightly charred on top, then transfer them to a bowl and season with salt & pepper.
3. Set up the bowl of tomatoes and the other ingredients at your grilling station for easy access.
4. Prepare a charcoal grill for grilling pizza creating a very hot fire on one side with couple of layers of charcoal just a few inches from the grate, and a low fire on the other side with just a few coals.
5. Oil the bottom of a baking sheet and place one of the dough balls on it.
6. Using your hands flatten it out to about 1/8 inch thick.
7. Carefully slide the dough onto the hot side of the grill.
8. Watch the dough closely making sure it doesn't burn by lifting it with tongs and peeking under.
9. When the dough is charred on the bottom, flip it over and brush it with olive oil.
10. Sprinkle with Parmigiano-Reggiano, dot with half of the fresh mozzarella and scatter half of the tomatoes on top.
11. With either the tongs or a spatula, slide the pizza to the cooler side of the grill.
12. Cover grill, making sure the vent holes are open, and cook for 3-5 more minutes.
13. The pizza is done when the cheese is melted and the bottom is slightly charred.
14. Using a spatula, remove pizza from grill to a serving tray, garnish with half of the basil, slice and serve.
15. Repeat with second ball of dough.