01 - Place chicken breasts between sheets of plastic wrap and pound to ¼-inch thickness using a meat mallet. Combine 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 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 pepper. Lay one slice of cheese and one slice of smoked ham (if using) on each chicken breast. Roll each breast up tightly, starting from the narrower end, 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, and the third with panko breadcrumbs mixed with paprika and dried dill. Dredge each chicken roll in flour, shaking off excess, then dip into the egg wash, and finally coat thoroughly with the panko mixture, pressing gently to ensure the crumbs adhere well on all sides.
05 - Arrange the breaded chicken rolls on the prepared baking sheet, seam side down. Spray the tops lightly with cooking spray or drizzle with a small amount of vegetable oil. Bake for 25 to 30 minutes, or until the exterior is deeply golden brown and the internal temperature registers 165°F on an instant-read thermometer.
06 - Remove from the oven and allow the chicken rolls to rest for 5 minutes so the juices redistribute. Carefully remove all toothpicks, slice each roll into pinwheel portions, and serve immediately while hot.