01 - Place chicken breasts between sheets of plastic wrap and pound to 1/4-inch thickness using a meat mallet. Combine pickle juice, garlic powder, and onion powder in a sealable bag or shallow dish. Add the pounded chicken, 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 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 bowls: the first with all-purpose flour, the second with beaten eggs mixed with milk or water, and the third with panko breadcrumbs combined with paprika and dried dill. Dredge each chicken roll in flour first, shaking off excess, then dip into the egg wash, and finally coat thoroughly with the panko mixture, pressing gently to ensure the breadcrumbs adhere evenly.
05 - Arrange the breaded rolls on the prepared baking sheet. 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 golden brown and the internal temperature registers 165°F on a meat thermometer.
06 - Allow the 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.