In a large bowl combine the warm water and yeast; let stand for 5 minutes. Add flour and 2 teaspoons salt, stir to combine. Turn out onto a lightly floured surface. Knead 6 to 8 minutes to make a moderately stiff dough that is smooth and elastic. Shape dough into a ball. Place in a lightly greased bowl; turn once to grease surface of dough. Cover with a kitchen towel. Let rise at warm room temperature 1 to 2 hours, until doubled in size.
For sauce, while dough is rising, mince and mash garlic with a large pinch of salt to make a paste. In a bowl stir together garlic paste, tomatoes, Parmesan, 1/4 cup oil, lemon peel, pepper, and remaining 1/2 teaspoon salt.
Position racks in the upper and lower thirds of an oven. Preheat oven to 425 degrees F.
Place dough on a well-floured work surface. Stretch or roll dough to a 22 x 14-inch rectangle. Scatter proscuitto and basil leaves on dough. Place tomato sauce along the lengthwise center of dough. Fold lower third of dough over tomato sauce, then fold over upper third of dough. Cut dough crosswise in 10 pieces. Transfer rolls, cut sides up, to two baking sheets, spacing about 2 inches apart. Gently press dough to expose some filling.
Bake rolls 25 to 30 minutes, rotating baking sheets between racks midway through baking time, until crusts are golden and hollow sounding. To serve, drizzle rolls with olive oil and sprinkle with snipped fresh basil.