In a medium bowl combine the water, yeast, and sugar. Stir briefly and set aside until the yeast begins to bubble and swell, about 5 minutes. Add 2½ cups flour, 3 tablespoons oil, and the salt. Stir until the dough holds together and then turn it out onto a lightly floured work surface. Knead the dough until it is smooth, 4 to 6 minutes. Shape the dough into a ball, put it in a lightly oiled bowl, and turn to coat with oil. Cover the bowl with plastic wrap and let stand in a warm place until the dough doubles in volume, 1 to 1½ hours.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Punch down the dough and turn it out onto a lightly floured work surface. Divide the dough into four equal pieces. Roll or press each piece into an approximate circle, 8 to 9 inches wide and about ¼ inch thick (don’t worry if it’s not exact; irregular shapes are fine). Oil four 12-inch squares of wax paper or parchment paper and place one dough round on each.
Drain the clams and reserve 1 tablespoon of the clam juice. In a small bowl whisk the ¼ cup oil, garlic, and the reserved 1 tablespoon clam juice. In another bowl combine the parsley, cheese, oregano, and lemon zest. Put the drained clams in a third bowl.
Brush the cooking grates clean. Place the dough on the cooking grate with the paper side facing up. Grill over direct medium heat, with the lid closed as much as possible, until the dough is well marked and firm on the underside, 2 to 4 minutes, rotating as needed for even cooking (don’t worry if the dough bubbles). Remove and discard the paper.
Turn the dough over and brush the tops with the oil mixture (be certain to use all of it). Distribute the minced garlic evenly over the dough. Working quickly, add the clams and then top with the parsley mixture. Close the lid and continue to cook until the cheese melts and the crusts are crisp, 2 to 5 minutes more. Remove the pizzas from the grill and season with salt and pepper. Cut into wedges and serve right away.