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 - If baking, preheat the oven to 400°F. Line a baking sheet with parchment paper and set aside.
03 - Remove the chicken breasts from the brine and pat them thoroughly 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 with a meat mallet to an even ¼-inch thickness.
04 - Layer one slice of cheese and one slice of ham (if using) on each pounded chicken breast. Sprinkle evenly with the chopped fresh herbs and season lightly with salt and freshly ground black pepper. Roll each breast up tightly, tucking in the sides to enclose the filling. Secure each roll with toothpicks to hold its shape.
05 - Arrange three shallow bowls side by side: the first with the all-purpose flour, the second with the beaten eggs, and the third with the panko breadcrumbs mixed with garlic powder and smoked paprika.
06 - Dredge each chicken roll in the flour, shaking off any excess. Dip into the beaten eggs, allowing excess to drip off, then coat thoroughly and evenly in the seasoned panko breadcrumb mixture, pressing gently to adhere.
07 - For frying: Heat the vegetable oil in a large skillet over medium heat. Fry the chicken rolls, turning carefully, until all sides are golden brown, about 3 to 4 minutes per side. Transfer to the prepared baking sheet and finish in the oven 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 generously with melted butter or oil, and bake for 25 to 30 minutes until golden brown and cooked through to 165°F internally.
08 - Allow the chicken rolls to rest for 5 minutes. Carefully remove the toothpicks, slice each roll into pinwheels, and serve immediately while hot.