Preheat oven to 350 degrees F. Lay the chicken breast on cutting board between 2 pieces of plastic wrap. Gently pound the chicken to 1/4-inch thickness with the flat side of a meat mallet. You don’t want “pock marks in it”. Using the ribbed side will possibly tear the chicken.
Remove the top sheet of plastic and lay 2 slices of ham on top of each piece of chicken. You want to cover the breast but not have overhang. Add about 2 tablespoon of grated gruyere over top of the ham leaving a ¼” edge of plain ham. Fold in the sides of the chicken over top of the ham and cheese and roll up tight like a jellyroll.
Use the plastic wrap and wrap each breast like a sausage forming a night tight log twisting the ends. Chill for 15-30 minutes.
Place the flour in a small pie pan adding in 1/2 teaspoon each salt and pepper. Place the bread crumbs in another pie pan adding in the thyme, garlic, remaining 1/2 teaspoon salt, remaining 1/2 teaspoon pepper, and melted butter. Use a spoon to mix all together. The bread crumbs will be *almost* wet.
Beat the egg and add to another bowl.
Unwrap the chicken and discard the plastic. Dredge the chicken roll in the flour. Dip it into the egg mixture and then place in the bread crumbs. You want to fully coat the roll.
Drizzle a baking pan with the oil and place the chicken on it, seam side down. Bake for 20 to 25 minutes until browned and cooked through.
*Note: if you want a crispier crust broil for 2-3 minutes.