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 - Preheat the oven to 400°F if baking the chicken rolls.
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 and opening it like a book. Place each butterflied breast between two sheets of plastic wrap and pound to ¼-inch thickness using a meat mallet.
04 - Layer one slice of cheese and one slice of ham (if using) on each pounded breast. Sprinkle with the chopped fresh herbs and season lightly with salt and pepper. Roll each breast up tightly, tucking in the sides, and secure with toothpicks.
05 - Place the flour in one shallow bowl, the beaten eggs in a second, and the panko breadcrumbs mixed with garlic powder and smoked paprika in a third.
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 the oil in a large skillet over medium heat. Fry the chicken rolls, turning every 3 to 4 minutes per side until golden brown on all sides. Transfer to a baking sheet and finish in the oven at 400°F for 15 minutes, or until the internal temperature reaches 165°F. For baking only: place the rolls on a parchment-lined baking sheet, drizzle with melted butter or oil, and bake for 25 to 30 minutes until golden and cooked through.
08 - Let the chicken rolls rest for 5 minutes. Remove the toothpicks, slice into pinwheels, and serve hot.