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, and place 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 Caper-Basil Vinaigrette. Garnish with fresh basil leaves and serve at room temperature.
Whisk together the vinegar, mustard, garlic, salt, and pepper in a medium bowl. Slowly whisk in the oil until emulsified. Stir in the capers and basil.