Preheat oven to 450 degrees F. In a small saucepan, bring 1 cup water and white wine vinegar to a simmer. Remove from heat and add mushrooms. Cover and let mushrooms sit, 15 minutes. Strain.
Slice bread horizontally to create 2 pizzas. Lightly oil a baking sheet and place pizzas crust side up. Brush top with 1/4 cup oil. In a bowl, squeeze tomatoes to break up slightly. Place on pizza halves. Pull pieces of Buffalo mozzarella from the ball and place on halves. Top with porcini mushrooms, shredded mozzarella, garlic and pepperoni. Bake until cheese has melted and crust is crispy, about 25 minutes. Remove from oven, drizzle with oil, sprinkle on Parmesan and spread with cut basil. Sprinkle with salt and pepper and serve.