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1
If using fresh tomatoes, cut an X (just through skin) in each, at end opposite stem, and immerse in boiling water for 10 seconds.
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2
Transfer to cold water with a slotted spoon, then peel.
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3
Seed and chop tomatoes (fresh or canned).
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4
Simmer tomatoes, salt, and 2 tablespoons oil in a 10-inch heavy skillet over moderate heat, stirring occasionally, until very thick and reduced to about 1 cup, 10 to 15 minutes.
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5
Transfer to a bowl to cool.
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6
Toss together mozzarella and remaining tablespoon oil.
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7
Do not punch down dough.
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8
Gently dredge 1 ball of dough in a bowl of flour to coat, then transfer to a lightly floured work surface.
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9
Holding 1 edge of floured dough in the air with both hands and letting bottom touch work surface, carefully move hands around edge of dough (like turning a steering wheel), allowing weight of dough to stretch round to roughly 7 inches in diameter.
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10
Lay dough round flat on floured surface and continue to stretch by pressing dough with your fingertips, working from center outward to edge, stretching it into a 9-inch round.
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11
Transfer to a large floured tray and make another round in same manner, then place it next to other round.
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12
Lightly rub a long sheet of plastic wrap with flour, then invert loosely over pizza rounds and let them stand to puff slightly while preparing grill, 10 to 20 minutes.
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13
Open vents on bottom of grill and on lid.
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14
Light a heaping chimneyful of charcoal and pour it evenly over 2 opposite sides of bottom rack (you will have a double or triple layer of charcoal), leaving clear a 9-inch-wide strip in middle (for grilling over indirect heat).
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15
Charcoal fire is medium-hot when you can hold your hand 5 inches above rack for 3 to 4 seconds.
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16
Preheat burners on high, covered, 10 minutes, then reduce heat to moderate.
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17
Remove plastic wrap from both rounds of dough and lightly brush dough with some oil.
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18
Carefully flip dough rounds, oiled sides down, with your hands onto middle of lightly oiled grill rack and brush top of each with oil.
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19
Grill crusts, uncovered, until undersides are golden brown (rotate them if 1 side of grill is hotter than the other), 2 to 3 minutes on gas grill or 4 to 6 minutes on charcoal grill.
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20
Flip crusts over with tongs and a spatula and top each crust with half of tomato sauce, spreading evenly over dough and leaving a 1/2-inch border around edge.
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21
Sprinkle mozzarella evenly over sauce and grill pizzas, covered with grill lid, until undersides are golden brown and cheese is melted, about 3 minutes on gas grill or 5 minutes on charcoal grill.
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22
Scatter basil over pizzas.
