Preheat your oven to 375°F (190°C).
Butterfly each chicken breast and layer with one slice of ham and one slice of Swiss cheese. Fold and secure with toothpicks.
Season the flour with salt and pepper. Dredge each stuffed breast in flour, dip in beaten eggs, and coat with breadcrumbs.
Heat olive oil in a skillet over medium heat and brown each chicken breast for 2–3 minutes per side. Transfer to a baking dish.
Bake in the oven for 25–30 minutes, or until the chicken is fully cooked.
While baking, melt butter in a saucepan over medium heat. Stir in heavy cream, chicken broth, Dijon mustard, and garlic powder. Simmer until thickened.
Pour the cream sauce over the baked chicken and serve hot.