1. Use a sharp knife to slice a chicken breast horizontally, starting at the thickest part and cutting almost to the other side (do not cut all the way through) until, when opened, breast lies flat. Place between two pieces of plastic wrap. Pound to an even 1/4 inch thick. Remove top layer of plastic; season with salt and pepper. Repeat with remaining chicken. Layer 1 1/2 ounces each of cheese and ham over surface of a breast; fold in half to enclose filling. Repeat with remaining chicken, cheese, and ham.

2. Preheat oven to 200 degrees. Place flour in a shallow dish; season with salt and pepper. Beat eggs with 2 tablespoons water in another dish; season with salt and pepper. Place panko in a third dish; season with salt and pepper. Heat 2 tablespoons oil in a large skillet over medium. Meanwhile, dredge a breast in flour, shaking off excess. Dip in eggs, turning to evenly coat. Dredge in panko, pressing gently to adhere. Transfer to a plate; repeat with remaining chicken. Add half the chicken to skillet and cook, turning once halfway through, until chicken is cooked through and golden brown, and cheese is melted, 14 to 16 minutes. Transfer to a wire rack set over a rimmed baking sheet to drain. Keep warm in oven. Cook remaining chicken with remaining 2 tablespoons oil. Serve immediately.