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1
For grilled pizza, start a medium-hot charcoal or wood fire or preheat a gas grill to the maximum.
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2
Roll or lightly press each dough ball into a flat round, lightly flouring the work surface and the dough as necessary (do not use more flour than you need to).
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3
Let the rounds sit for a few minutes, then roll or pat out the dough, as thinly as you like, turning occasionally and sprinkling the top with flour as necessary.
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4
For baked pizza, preheat the oven to 500F.
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5
Oil one or more baking sheets, then press each dough ball into a flat round directly on the oiled sheet(s).
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6
Then pat out the dough, as thinly as you like, oiling your hands if necessary.
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7
If your oven is equipped with a baking stone, roll or pat out the dough as for grilled pizza, putting it on a peel to transfer it to the oven.
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8
Meanwhile, thinly slice the tomatoes.
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9
Salt the slices lightly and let them sit for at least 20 minutes, then drain off any accumulated liquid.
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10
To grill the pizza, slide it directly onto the grill.
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11
Cook until brown grill marks appear, 3 to 5 minutes, depending on your grill heat.
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12
Turn with a spatula or tongs, then top with the tomato, Parmigiano-Reggiano, and basil.
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13
Cover the grill and cook until the bottom is crisp and brown and the top hot.
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14
To bake the pizza, top with the tomato and Parmigiano-Reggiano, slide the baking sheet into the oven (or the pizza itself onto the stone), and bake for about 10 minutes, depending on the oven heat, or until nearly done.
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15
Sprinkle with the basil and cook until the pizza is nicely browned.
