Preheat oven to 180 degrees C. Coat a 22cm x 32cm baking dish with cooking spray.
Place 3 schnitzels side by side in the prepared pan and season with salt and pepper. Top with overlapping ham slices, starting 5mm away from the edges of the schnitzels. 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 stock around the edges. Top with the remaining 1/2 cup Cheddar cheese.
Bake in the preheated oven until chicken is no longer pink in the center and the juices run clear, 30 to 35 minutes.
Turn oven grill on to high and cook until top is crispy and golden brown, about 5 minutes more.