Preheat the oven to 350°F (175°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.
Season both sides of the chicken breasts with salt, black pepper, garlic powder, and dried thyme.
On each flattened chicken breast, place one slice of ham and one slice of Swiss cheese.
Carefully roll up each chicken breast tightly, securing with toothpicks if needed.
Place the flour in a shallow dish.
In a separate shallow dish, whisk together the eggs and water.
In a third shallow dish, combine the breadcrumbs with a pinch of salt and pepper.
Dredge each chicken roll first in the flour, shaking off excess, then dip into the egg mixture, and finally coat evenly with the breadcrumbs.
In a large oven-safe skillet, heat the butter and olive oil over medium heat until hot.
Add the chicken rolls and cook for about 3-4 minutes on each side, until golden brown.
Transfer the skillet to the preheated oven and bake for 15-20 minutes, or until the chicken is cooked through and the internal temperature reaches 165°F (74°C).
While the chicken bakes, prepare the sauce: In a small saucepan, combine the chicken broth, heavy cream, and Dijon mustard.
Bring the sauce to a simmer over medium heat, stirring frequently, until slightly thickened, about 5 minutes.
Remove the toothpicks from the chicken rolls before serving.
Serve the chicken cordon bleu hot, drizzled with the creamy Dijon sauce.