Preheat the oven to 375°F (190°C).
Place each chicken breast between two sheets of plastic wrap and gently pound with a meat mallet or rolling pin until about 1/4 inch thick.
Lay out the beef slices flat on a clean surface. Place one pounded chicken breast on top of each beef slice.
On each chicken breast, layer two slices of ham and two slices of Swiss cheese evenly.
Carefully roll up each layered chicken and beef slice tightly, tucking in the sides to enclose the filling completely. Secure with toothpicks if needed.
In a shallow bowl, place the all-purpose flour.
In a second shallow bowl, whisk together the eggs and water until combined.
In a third shallow bowl, mix the seasoned bread crumbs, garlic powder, onion powder, black pepper, and salt.
Dredge each rolled cordon bleu first in the flour, shaking off excess, then dip into the egg mixture, and finally coat thoroughly with the bread crumb mixture.
In a large oven-safe skillet, heat the olive oil and butter over medium heat until melted and hot.
Add the breaded cordon bleu 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 oven and let rest for 5 minutes before removing toothpicks and slicing to serve.