In a small bowl, whisk together both vinegars, the garlic, parsley, Italian seasoning, Dijon mustard and dry mustard. While whisking, add oil in a thin stream. Whisk in salt and pepper and set aside.
In a large bowl, combine flour and Cajun seasoning. Peel off tough outer skin from onion and thinly slice onion. Add slices to bowl and toss to coat. Heat oil in a medium-size saucepan to 365° on a deep-fat thermometer. Add onion in batches (about 3/4 cup per batch), being careful that oil doesn't overflow. Cook onion about 1 minute, then transfer to paper towel-lined tray to cool. Repeat with all the onion (about 4 batches total).
Grill or broil chicken breasts, turning once, until cooked through, about 12 minutes. Heat barbecue sauce in microwave, about 30 seconds. Cut chicken into 1/2-inch pieces and stir into barbecue sauce.
Place romaine, cucumber, tomato and mushrooms on a large cutting board (or you can work in batches). Chop vegetables until they are 1- to 1-1/2-inch pieces. Place lettuce mixture in center of large platter and toss with 2/3 cup of the Balsamic Dressing (reserve remainder for another use). Top with fried onions and mozzarella cheese. Drizzle with ricotta cheese and Parmesan. Spoon barbecue chicken around edge of platter. Garnish with pepperoncini and serve.