Toss the chicken, bell peppers, onion, parsley, almonds, and capers in a large bowl with enough vinaigrette to moisten. Season the chicken salad with salt and pepper, to taste.
Arrange 1 large lettuce cup and 1 small lettuce cup on each plate, overlapping slightly. Spoon the chicken salad into the lettuce cups. Drizzle more vinaigrette over the salads. Sprinkle with the Parmesan and serve.
Do-Ahead Tip: The chicken salad can be prepared 4 hours ahead and refrigerated. Just before serving, spoon the salad into the lettuce cups. 1/2 cup red wine vinegar 1/4 cup lemon juice 2 teaspoons honey 2 teaspoons salt Freshly ground black pepper 1 cup olive oil
Combine the vinegar, lemon juice, honey, salt, and pepper in a blender. With the machine running, gradually add the oil and blend until emulsified.