Melt the butter in a small skillet over medium heat. Add the panko breadcrumbs and cook, stirring constantly, until they are a deep, golden brown. Remove from the heat and stir in a dash of salt. Set aside.

Preheat the oven broiler to high. Line a rimmed baking sheet with foil.

Place the chicken breasts on a cutting board, and using a sharp knife, cut the breasts in half lengthwise, forming 4 thinner pieces. (Alternately, you can use 4 chicken cutlets.) Season both sides of the chicken with the remaining salt and the pepper.

Heat the vegetable oil in a large skillet over medium heat. Add the chicken and cook until browned, 3-4 minutes. Flip the chicken over and cook the second side until browned. (You may need to do this in 2 batches if your skillet isn’t big enough.) Transfer the chicken to the prepared baking sheet. Top each piece of chicken with some of the ham and a slice of cheese.

Broil the chicken until the cheese is melted and bubbly, 3-4 minutes. Make sure to keep an eye on it as it goes quickly.

Serve the chicken sprinkled with the reserved breadcrumbs.