01 - In a large bowl or resealable plastic bag, combine the dill pickle juice, sugar, and black peppercorns. Add the chicken breasts, seal or cover, and refrigerate for at least 1 hour and up to 4 hours.
02 - If baking, preheat the oven to 400°F. Line a baking sheet with parchment paper.
03 - Remove the chicken from the brine and pat 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 - Lay one slice of cheese and one slice of ham (if using) on each pounded breast. Sprinkle with chopped herbs and season lightly with salt and pepper. Roll each breast up tightly, tucking in the sides, and secure with toothpicks.
05 - Arrange three shallow bowls: one with the flour, one with the beaten eggs, and the third with panko breadcrumbs mixed with garlic powder and smoked paprika.
06 - Dredge each chicken roll in the flour, shaking off excess. Dip into the beaten eggs, then coat thoroughly in the seasoned panko mixture, pressing gently to adhere.
07 - For frying: Heat oil in a large skillet over medium heat. Fry the chicken rolls, turning occasionally, until all sides are golden brown, about 3–4 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 rolls on the parchment-lined baking sheet, drizzle with melted butter or oil, and bake for 25–30 minutes until golden brown and cooked through to 165°F.
08 - Let the chicken rolls rest for 5 minutes. Remove toothpicks, slice into pinwheels, and serve hot.