Put shredded chicken in a mixing bowl and season to taste with salt and pepper. Toss to distribute seasoning. Add olive oil, vinegar, Italian seasoning, giardiniera, and parmesan. Set aside.
Preheat the oven to 350°. Cut the rolls open, leaving a hinge. Open slightly and arrange the cheese over the cut “valley”. Bake until the cheese is melted and the roll is beginning to crisp.
Divide the chicken filling between the rolls. Serve immediately or wrap and store in the fridge until needed.