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 large sealable bag or shallow dish. Add the flattened 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 pepper. Lay one slice of cheese and one slice of smoked ham, if using, on top of each breast. Roll each breast up tightly from one end to the other and secure the seam with toothpicks to hold the filling inside.
04 - Set up three shallow dishes: the first with flour, the second with eggs beaten together with milk, 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 mixture until fully coated, then roll in the panko mixture, pressing gently so the crumbs adhere evenly on all sides.
05 - Arrange the breaded rolls on the prepared baking sheet. Spray the tops lightly with cooking spray or drizzle with a thin stream of vegetable oil. Bake for 25 to 30 minutes until the exterior is golden brown and the internal temperature registers 165°F at the thickest part.
06 - Remove from the oven and let the rolls rest for 5 minutes to allow the juices to redistribute. Carefully remove the toothpicks, slice each roll into pinwheel portions, and serve immediately while hot.