Heat oven to 350 degrees F. Place one breast half between two sheets of waxed paper on a cutting board. Pound to an even thickness of 1/4 inch with the flat side of a meat mallet, a rolling pin or cast iron pan. Repeat with remaining chicken breasts.

Spread out one piece of ham; pile 1/4 cup of the cheese in the middle, then tightly fold up burrito-style to enclose. Spread one pounded chicken breast with 2 teaspoons of the mustard. Place ham bundle on chicken; slide to one narrow end. Roll chicken breast, enclosing bundle. Secure with a toothpick or two; tuck in any loose ends. Repeat for a total of 4 stuffed chicken breasts.

Spread flour onto a dinner plate, and pour eggs into a pie dish and bread crumbs onto a second dinner plate. Coat one stuffed breast with flour, dip in egg completely, then coat with bread crumbs, adhering with fingers if needed. Repeat with all chicken pieces.

Heat oil in a 12-inch skillet with a tight-fitting lid over medium-high heat. Add chicken pieces and brown, covered, 3 minutes. Flip chicken over, and continue to cook, covered, an additional 2 to 3 minutes. Transfer to a shallow baking dish and bake at 350 degrees F for 12 to 15 minutes or until temperature registers 160 degrees F on an instant-read thermometer.