Prepare the grill for direct heat: If using a charcoal grill, light the charcoal or wood briquettes; when the briquettes are ashen, distribute them evenly over the cooking area. If using a gas grill, preheat by setting all burners on high and closing the lid for 10 minutes; then turn off one of the burners on a two-burner grill or the two end burners on a three-burner grill and reduce the heat to medium, or 350 degrees. When the crust goes on the grill, the heat should be medium; you should be able to hold your hand about 6 inches above the coals or grate for about 7 seconds. Have ready a spray water bottle for taming any flames. Brush the grill grate.
Meanwhile, combine the garlic and oil in a small bowl.
Lightly flour a work surface. Roll out and shape the dough to a round approximately 12 to 14 inches across with a thickness of about 1/8 inch and no more than 1/4 inch; the edges should be closer to 1/2 inch thick.
Sprinkle a little flour and cornmeal (as needed) on the pizza peel, then transfer the dough to it. Use the peel to slide the dough onto the grill grate. Close the lid and cook for 6 to 9 minutes or until the underside is browned and perhaps blackened in spots; check the underside after 5 minutes and proceed as needed.
Use tongs to transfer the crust to a platter; invert the crust. Immediately brush its grilled side with the garlic-oil mixture. Scatter the clams, clam juice, oregano and cheese evenly over the surface. Return the pizza to the grill, close the lid and cook for 5 to 6 minutes, until the crust is golden brown and crisp.
Transfer to a plate; cut and serve right away.