Preheat the oven to 375°F (190°C).
Lay the beef slices flat on a clean surface. Place one chicken breast half on each beef slice.
Top each chicken breast with 2 slices of ham and 2 slices of Swiss cheese.
Carefully roll up each beef and chicken stack tightly, securing with toothpicks if needed.
In a shallow bowl, combine the flour, salt, black pepper, garlic powder, and paprika.
In a second shallow bowl, whisk together the eggs and water.
Place the breadcrumbs in a third shallow bowl.
Dredge each beef 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 heat.
Add the rolls 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 serving. Remove toothpicks before serving.