Heat your grill to high. Brush bell peppers with 2 tablespoons of the oil and season with 1/2 teaspoon of salt and 1/2 teaspoon of pepper; grill the peppers until charred on all sides. Remove from the grill, place in a bowl, cover with plastic wrap and let sit for 10 minutes. Remove the skin from the peppers, brush them with the remaining oil and season with salt and pepper. Place them back on the grill and cook until slightly charred on all sides.
Slice peppers into quarters or to fit over bread slices. Place a slice of mozzarella over each pepper and drizzle with the Basil Vinaigrette. Garnish with fresh basil leaves and serve at room temperature.