01 - Place chicken breasts between sheets of plastic wrap and pound to ¼-inch thickness using a meat mallet. Combine dill pickle juice, garlic powder, and onion powder in a large sealable bag or shallow dish. Add the pounded chicken breasts, seal tightly, and refrigerate for at least 1 hour or up to overnight for deeper flavor penetration.
02 - Preheat the oven to 400°F. Line a baking sheet with parchment paper and lightly coat the surface with cooking spray to prevent sticking.
03 - Remove chicken from the brine and pat thoroughly dry with paper towels. Season each breast lightly with salt and freshly ground black pepper. Lay one slice of cheese and one slice of smoked ham (if using) on each chicken breast. Roll each breast up tightly from one end to the other and secure with toothpicks to hold the shape during baking.
04 - Set up a standard breading station with three shallow dishes: the first with all-purpose flour, the second with beaten eggs whisked together with milk or water, and the third with panko breadcrumbs mixed with paprika and dried dill. Dredge each chicken roll in flour, shaking off excess. Dip into the egg wash, allowing excess to drip off. Press firmly into the panko mixture, coating evenly on all sides.
05 - Arrange the breaded chicken rolls on the prepared baking sheet, seam side down. Spray the tops generously with cooking spray or drizzle lightly with vegetable oil. Bake for 25 to 30 minutes until the exterior is deeply golden brown and the internal temperature registers 165°F on an instant-read thermometer.
06 - Allow the chicken rolls to rest for 5 minutes after removing from the oven. Carefully remove all toothpicks, slice each roll into pinwheel portions, and serve immediately while hot.