1. Make sauce and prepare cheese: If using fresh tomatoes, cut an X (just through skin) in each, at end opposite stem, and immerse in boiling water for 10 seconds. Transfer to cold water with a slotted spoon, then peel.
2. Seed and chop tomatoes (fresh or canned).
3. 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. Transfer to a bowl to cool.
4. Toss together mozzarella and remaining tablespoon oil.
5. Form 2 pizza rounds: Do not punch down dough. Gently dredge 1 ball of dough in a bowl of flour to coat, then transfer to a lightly floured work surface. 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. 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. Transfer to a large floured tray and make another round in same manner, then place it next to other round. 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.
6. To prepare charcoal grill: Open vents on bottom of grill and on lid. 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).
7. Charcoal fire is medium-hot when you can hold your hand 5 inches above rack for 3 to 4 seconds.
8. To prepare gas grill: Preheat burners on high, covered, 10 minutes, then reduce heat to moderate.
9. To grill pizzas by either method: Remove plastic wrap from both rounds of dough and lightly brush dough with some oil. Carefully flip dough rounds, oiled sides down, with your hands onto middle of lightly oiled grill rack and brush top of each with oil. 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.
10. 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. 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.
11. Scatter basil over pizzas.

Yield: 4 servings