Preheat oven to 350 degrees F (175 degrees C). Coat a 9x13-inch baking pan with cooking spray.

Place 3 cutlets side by side in the prepared pan and season with salt and pepper. Top with overlapping ham slices, starting 1/4 inch away from the edges of the cutlets. Sprinkle 1 1/2 cups Cheddar cheese over ham. Lay remaining 3 cutlets side by side on top, covering all the Cheddar cheese. Sprinkle flour on top, spreading with hands to cover.

Combine beaten egg and milk in a bowl; pour over the chicken.

Combine bread crumbs with oregano in a bowl; sprinkle over the top and sides to cover all surfaces of chicken. Spoon any excess egg mixture from the bottom of the pan over the chicken. Drizzle melted butter on top; pour chicken broth around the edges. Top with the remaining 1/2 cup Cheddar cheese.

Bake in the preheated oven until cutlets are no longer pink in the center and the juices run clear, 30 to 35 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).