Pound the chicken breasts to an even thickness, about 1/2 inch, between two sheets of plastic wrap. This ensures even cooking and a tender bite.
Season the chicken lightly with salt and pepper. Spread a thin layer of Dijon mustard on each breast to add a subtle zing that complements the ham and cheese.
Layer a slice of ham and a slice of cheese on each breast, then roll each breast up tightly like a jelly roll. Secure with toothpicks if needed.
Set up a breading station: place flour in one shallow dish, beaten eggs in another, and a mixture of breadcrumbs and Parmesan in a third. Lightly season the breadcrumb mixture with a pinch of salt.
Dredge each chicken roll in flour, shake off excess, dip into the egg, then roll in the breadcrumb-Parmesan mixture until fully coated.
Heat olive oil in a skillet over medium-high heat. Sear the chicken rolls on all sides until golden brown, about 2–3 minutes per side.
Transfer the browned rolls to a baking dish. Dot with small bits of butter on top for extra richness.
Bake in a preheated 375°F (190°C) oven for 20–25 minutes, or until the chicken reaches an internal temperature of 165°F (74°C) and the cheese is melty.
Let the roulades rest for 5 minutes before slicing. Remove toothpicks, slice crosswise, and serve with a simple pan sauce or a light mushroom cream sauce if you like.