Place one chicken breast between two sheets of plastic wrap or in a large zip-top bag. Using a meat mallet or heavy skillet, gently pound the thickest part of the breast until the breast is an even 1/4-inch thickness. Repeat with remaining breasts.

Season

Season both sides of each flattened breast with the measured salt, black pepper, and garlic powder, dividing evenly.

Assemble filling

On each flattened breast, place two slices of ham so they cover most of the surface, then top with two slices of cheese. Leave a 1/2-inch border around edges to make sealing easier.

Roll and secure

Starting from one short end, carefully roll each breast tightly around the ham and cheese to make a neat cylinder. Tuck in the ends as you roll to enclose the filling. If needed, secure each roll with 2 toothpicks placed lengthwise through the seam so rolls don't open while cooking.

Breading Station

Prepare breading

Set up three shallow dishes: in the first, place the flour; in the second, whisk together the eggs and water (or milk) until homogenous; in the third, mix the breadcrumbs, grated Parmesan (if using), paprika, and the 1/4 tsp salt.

Dredge chicken

One at a time, dredge each stuffed chicken roll in flour, shaking off excess, then dip into the egg wash, letting excess drip off, and finally press into the breadcrumb mixture to coat thoroughly. For extra-crispy, repeat the egg and breadcrumb steps for a double coating. Place each breaded roll on a plate and let rest 5–10 minutes for the coating to adhere.

Cooking & Finishing

Pan-fry

In a large ovenproof skillet, heat the oil and butter together over medium heat until the butter stops foaming and the fat is hot but not smoking. Add the breaded chicken rolls seam-side down and cook 3–4 minutes without moving, until golden brown. Turn to brown the other sides, about 2–3 minutes per side so all sides are golden (total browning time ~8–10 minutes).

Finish in oven

Preheat the oven to 375°F (190°C) (if not preheated). Transfer the skillet with the browned rolls to the oven and bake 12–18 minutes, until the internal temperature of the thickest part registers 160–165°F (71–74°C) on an instant-read thermometer, and juices run clear. (If you used toothpicks, keep them in while baking.)

Rest and remove toothpicks

Remove the skillet from the oven. Transfer the rolls to a cutting board and let rest 5 minutes. Remove toothpicks before slicing. Resting lets the cheese settle and prevents it from immediately spilling out.

Slice and serve

Slice each roll on the bias into 1-inch thick rounds or serve whole. Spoon any pan juices over the top, sprinkle with chopped parsley if using, and serve with lemon wedges for squeezing.