Combine fish sauce, ground pepper, lemon or calamansi juice and kosher salt in a small bowl.
Pound chicken breasts to ½ inch thickness. Marinade the chicken with the fish sauce mixture for at least an hour. Remove excess liquid by squeezing gently.
Place the flour, egg, and breadcrumbs in three separate deep plates. Add a teaspoon of salt and ¼ teaspoon of ground pepper to the breadcrumbs and mix well.
Place two plastic wrap on a flat surface and lay a chicken breast fillet on top of each wrapper. Top each half with ham and cheese. Roll up each breast fillet jelly-roll fashion by pushing the plastic wrap underneath from one side to the other like sushi. Try to make it as compact as possible.
Remove plastic and roll each in flour covering the chicken completely. Then dip each roll in egg and lastly dredge in breadcrumb mixture, completely covering the chicken and the ends filled. Cover each roll tightly with a plastic or foil wrap. Place the rolls in the fridge overnight.
In a deep pan or pot over medium heat, heat enough oil to deep fry the chicken. Deep fry the rolls until cooked.
Remove from oil and transfer to a plate lined with paper towels to remove excess oil.
Cut into slices and serve.