01 - In a large bowl or resealable plastic bag, combine the dill pickle juice, granulated sugar, and whole black peppercorns. Add the chicken breasts, seal or cover, and refrigerate for at least 1 hour or up to 4 hours.
02 - Preheat the oven to 400°F. Line a baking sheet with parchment paper if baking.
03 - Remove the chicken breasts from the brine and pat them dry with paper towels. Butterfly each breast by slicing horizontally almost all the way through, then opening it like a book. Place each butterflied breast between two sheets of plastic wrap and pound to an even ¼-inch thickness using a meat mallet or rolling pin.
04 - Layer one slice of cheese, one slice of ham if using, and a generous sprinkle of chopped fresh herbs onto each pounded chicken breast. Season lightly with salt and freshly ground black pepper. Roll each breast up tightly, tucking in the sides, and secure with toothpicks to hold their shape.
05 - Arrange three shallow bowls: the first with all-purpose flour, the second with beaten eggs, and the third with panko breadcrumbs mixed with garlic powder and smoked paprika.
06 - Dredge each rolled chicken breast first in the flour, shaking off any excess, then dip into the beaten eggs, and finally coat thoroughly in the seasoned panko breadcrumbs, pressing gently to adhere.
07 - For frying: Heat vegetable oil in a large skillet over medium heat. Fry the chicken rolls, turning occasionally, until golden brown on all sides, about 3 to 4 minutes per side. Transfer to the prepared baking sheet and bake for 15 minutes, or until the internal temperature reaches 165°F. For baking only: Arrange the breaded rolls on the parchment-lined baking sheet, drizzle with melted butter or oil, and bake for 25 to 30 minutes until golden brown and cooked through.
08 - Allow the chicken rolls to rest for 5 minutes before slicing. Remove toothpicks, slice into pinwheels, and serve hot.