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, seal or cover, 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.
03 - Remove chicken from the brine and pat thoroughly dry with paper towels. Season each piece 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.
04 - Set up a standard breading station with three shallow bowls: the first with 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 the flour, shaking off excess, then dip into the egg wash, and finally press firmly into the panko mixture to ensure an even, adherent coating.
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 deeply golden and the internal temperature reaches 165°F when checked with 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.