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 and 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 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, 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 the shape.
05 - Arrange three shallow bowls side by side. Place the all-purpose flour in the first bowl, the beaten eggs in the second, and the panko breadcrumbs mixed with garlic powder and smoked paprika in the third.
06 - Dredge each chicken roll in the flour, shaking off any excess. Dip into the beaten eggs, coating completely, then roll in the seasoned panko breadcrumbs, pressing gently to ensure an even, thorough coating on all sides.
07 - For the pan-fried method, heat 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 bake for 15 minutes, or until the internal temperature reaches 165°F. For the oven-only method, arrange the 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 to 165°F internally.
08 - Allow the chicken rolls to rest for 5 minutes before slicing. Remove the toothpicks, cut into pinwheel slices, and serve hot.