Place each chicken breast between two sheets of plastic wrap or parchment paper. Use a meat mallet or rolling pin to gently pound the chicken to an even thickness (about 1/4 inch thick). Season both sides of each chicken breast with salt and pepper.
Lay one slice of ham and one slice of cheese on each flattened chicken breast. Roll up the chicken breast tightly, starting from the narrower end, to enclose the ham and cheese completely. Secure the rolls with toothpicks to prevent them from unraveling during cooking. Or you can use cling wrap to keep it in place and let it stay in the fridge for 30 minutes.
Set up three shallow dishes: one with flour, one with beaten eggs, and one with breadcrumbs (you can mix in garlic powder, onion powder, salt, and pepper for extra flavor in the flour). Dredge each chicken roll in the flour, making sure it’s fully coated. Shake off any excess. Dip the floured chicken roll into the beaten eggs, ensuring it’s fully coated. Finally, roll the chicken in the breadcrumbs, pressing down to adhere the breadcrumbs to the chicken.
Heat cooking oil in a pan enough to submerge half of the chicken rolls. Fry the chicken rolls on all sides until golden brown (about 3 minutes per side).
Let it cool in a cooling rack before slicing it.