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 two slices of beef flat on a clean surface. Place one flattened chicken breast on top of the beef slices.
On top of the chicken breast, layer two slices of ham and two slices of Swiss cheese evenly.
Carefully roll the chicken breast, ham, and cheese inside the beef slices, tucking in the sides to form a tight roll. Secure with toothpicks if necessary.
In a shallow bowl, place the flour. In a second bowl, whisk together the eggs and water. In a third bowl, combine the breadcrumbs, salt, black pepper, garlic powder, and paprika.
Dredge each beef-wrapped chicken 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 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 the toothpicks before serving. Let rest for 5 minutes, then slice and serve warm.