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 two slices of beef flat on a clean surface for each chicken breast, slightly overlapping to form a rectangle large enough to wrap the chicken.
Place one pounded chicken breast on top of the beef slices.
Top each chicken breast with two slices of ham and two slices of Swiss cheese.
Carefully roll the chicken breast tightly inside the beef and ham, securing the roll with toothpicks if needed.
In a shallow dish, combine the flour, salt, black pepper, garlic powder, and paprika.
In a second shallow dish, whisk together the eggs and water.
Place the breadcrumbs in a third shallow dish.
Dredge each beef-wrapped chicken roll first in the seasoned flour, then dip into the egg mixture, and finally coat evenly with breadcrumbs.
In a large oven-safe skillet, heat the olive oil and butter over medium-high heat.
Add the breaded chicken rolls and sear on all sides until golden brown, about 2-3 minutes per side.
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 to serve.