Prepare gas grill for cooking over moderate heat. Preheat burners on high, covered, 10 minutes, then, if necessary, reduce to heat specified in recipe.
Meanwhile, mince garlic with a large heavy knife, then mash to a paste with a pinch of salt using flat side of knife. Transfer to a small bowl and stir in 1 teaspoon olive oil.
Stretch dough with your fingers into a 12- by 10-inch rectangle on a large baking sheet and rub with 1 teaspoon olive oil. Cover dough with plastic wrap.
Toss zucchini with salt (1/4 teaspoon) and 1 tablespoon olive oil in a bowl. Grill zucchini on lightly oiled grill rack, covered, turning occasionally, until just tender, 4 to 5 minutes total. Return to bowl.
Bring dough, garlic oil, and cheese to grill area. Grill dough, oiled side down, on lightly oiled grill rack, covered, until underside is golden brown, 2 to 4 minutes.
Turn crust over using tongs and brush with garlic oil. Sprinkle with half of cheese, then arrange zucchini, overlapping slightly, in 1 layer on cheese. Sprinkle remaining cheese on top and grill pizza, covered, until underside is golden brown and cheese is melted, 3 to 5 minutes. Transfer pizza with tongs and a spatula to a cutting board, then sprinkle with oregano and drizzle with remaining teaspoon olive oil.
