Preheat oven to 450°F with rack in lowest position. Place baking stone in oven while pre-heating. Let stone heat up for 30 minutes while preparing pizza dough and toppings.
Roll dough to a 12-14 inch wide circle on a floured surface.
Meanwhile, heat 1 tablespoon oil in a large skillet over medium-high. Add onions and cook, stirring occasionally, until softened, about 8 minutes. Remove from heat; stir in vinegar and pepper.
When stone is hot pull out of over. Place pizza dough on hot stone.
Scatter onions, mozzarella, and garlic over crust. Return to oven and bake until cheese is melted and crust is brown, 8 to 10 minutes.
Spoon tapenade evenly over pizza and sprinkle with basil, cut & serve