Pound each chicken breast between sheets of plastic wrap to an even 1/4-inch thickness and season with the salt and pepper.
Place one ham slice and one cheese baton on each breast, roll tightly and secure the seam with toothpicks.
Chill the rolls for 20 minutes, then coat each in flour, beaten eggs mixed with water and panko.
Heat the oil in an oven-safe skillet and brown the rolls over medium-high heat for 2 minutes per side.
Bake at 375°F for 18 to 22 minutes until the chicken reaches 165°F, rest for 5 minutes and remove every toothpick before slicing.