In a 10- to 12-inch saute pan heat olive oil until smoking. Add onion and cook until soft, about 9 to 10 minutes. Meanwhile, cut eggplant into 1/4-inch thick rounds. When onion has softened, add eggplant to pan and cook, stirring regularly until eggplant has darkened and is soft. Add 2 tablespoons red wine vinegar and remove from heat to cool. Cut fresh mozzarella into 1/4-inch cubes and add to cooled eggplant mixture. Add basil and season to taste with salt and pepper.
Grill or toast bread on both sides and spoon generous amount over each slice of bread. Place 3 slices Bresaola over top of eggplant mixture and serve.