makes one 12-inch pizza
Add the tomatoes (with their juices) to the bowl of a food processor, and pulse several times to crush the tomatoes. Transfer the tomatoes to a fine-mesh strainer set over a medium bowl and allow to drain 20-30 minutes, stirring occasionally. Combine the drained tomatoes with sugar, 2 Tablespoons of basil, garlic, and 1/8 teaspoon salt.
Meanwhile, preheat the grill to high heat (450-500 degrees) for at least 15-20 minutes.
As the tomatoes drain, shape the pizza on a well-floured surface by flattening the dough into an 8-inch disk, pressing to eliminate air pockets. Stretch the outer edges with flattened fingers while giving quarter turns until the dough is a 12-inch circle. Allow to sit at room tempeature 10-15 minutes. Using a pastry brush, oil the dough well.
Using a cutting board or pizza peel, transfer pizza, oil-side down, onto the grates of the grill. Cook 5-10 minutes, or until under-side of dough is golden-brown. Brush the top with more oil. Using 2 spatulas, carefully flip the dough. Working quickly, spread the tomatoes on the now-cooked side, then scatter the cheese evenly on top. Close the grill cover and continue too cook until the crust is cooked through and the cheese has melted. Carefully slide pizza onto a platter, and sprinkle with remaining basil, 1 teaspoon of olive oil, and a pinch of kosher or sea salt. Serve immediately.