Preheat the oven to 375°F (190°C).
Place each chicken breast between two sheets of plastic wrap and pound with a meat mallet or rolling pin until about 1/4 inch thick.
Lay out the beef slices on a clean surface. Place one pounded chicken breast on top of two beef slices, overlapping slightly to cover the chicken.
On top of the chicken, layer 2 slices of ham and 2 slices of Swiss cheese evenly.
Carefully roll the beef and chicken around the filling, tucking in the sides to form a tight roll. Secure with toothpicks if necessary.
In a shallow dish, place the flour.
In a second shallow dish, whisk together the eggs and water until combined.
In a third shallow dish, combine the breadcrumbs, garlic powder, Italian seasoning, salt, and black pepper.
Dredge each beef chicken cordon bleu roll first in the flour, shaking off excess, then dip into the egg mixture, and finally coat evenly with the breadcrumb mixture.
In a large oven-safe skillet, heat the olive oil and butter over medium heat until melted and hot.
Add the breaded rolls to the skillet and cook for 3-4 minutes on each side, until golden brown.
Transfer the skillet to the preheated oven and bake for 20-25 minutes, or until the chicken is cooked through and the internal temperature reaches 165°F (74°C).
Remove from the oven and let rest for 5 minutes before removing toothpicks and slicing.
Serve warm.