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 your 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 deli ham (if using), and a generous sprinkle of chopped fresh herbs on each pounded chicken breast. Season lightly with salt and freshly ground black pepper.
05 - Roll each filled chicken breast up tightly, tucking in the sides as you go. Secure the seam with toothpicks to hold the rolls together during cooking.
06 - 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.
07 - 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.
08 - For frying: Heat vegetable oil in a large skillet over medium heat. Fry the chicken rolls, turning every 3 to 4 minutes, until all sides are golden brown. 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 to 30 minutes until golden and cooked through.
09 - Allow the chicken rolls to rest for 5 minutes before slicing. Remove toothpicks, slice into pinwheels, and serve hot.