Brush pizza dough with olive oil and dust with salt and pepper. Place on a medium indirect grill, covered, for approximately 3 minutes. Flip dough and grill in the same manner for 1 minute.
Remove dough from the grill, brush with the pesto, then the pizza sauce, stopping a 1/2- inch before the edge of the crust. Spread 1/2 of the mozzarella and Parmesan on the sauced surface. Lay out sliced tomato. Dust with basil. Top with remaining mozzarella and the parmesan. Bake on grill over indirect heat for approximately 4 minutes.
Remove from the grill and sprinkle with green onions. Slice into 8 pieces.