For the Chicken

1. Slice and Pound the Chicken: Cut the chicken breasts in half horizontally to form flat fillets. Place a fillet in a resealable freezer bag and use a meat mallet or rolling pin to pound it to an even thickness between ⅛ and ¼ inch. Remove the pounded cutlet and set it aside. Repeat with the remaining three chicken pieces.
2. Prepare the Chicken for Filling: Lay each chicken breast vertically, smooth side down, on a 12x16-inch piece of plastic wrap that is positioned horizontally (looking down at the plastic wrap, it should be wider than tall). Evenly season the rough sides (facing up) with ½ teaspoon salt and ½ teaspoon pepper (you'll use about ⅛ teaspoon salt and pepper for each piece of chicken).
3. Create the Ham and Cheese Packets: Place ¼ cup shredded Gruyère in the center of each ham slice. Fold the ham over the cheese, starting with the sides and then the top and bottom. This encases the cheese within the ham, forming neat packets. Place the packets seam-side down to keep them closed.
4. Assemble and Roll: Place these ham and cheese packets atop the seasoned chicken breasts, towards the end closest to you. Begin rolling the chicken away from you, enclosing the ham and cheese packet tightly. As you roll, tuck in the chicken's sides. Once rolled, wrap the chicken tightly in the plastic wrap, squeezing gently to seal and twisting both ends to form a log-like shape. Repeat with the remaining pieces. Refrigerate the chicken for at least 30 minutes or up to 1 day.
5. Preheat the oven to 375°F and set an oven rack in the middle position. Line a rimmed baking sheet with aluminum foil and spray with nonstick cooking spray.
6. Bread the Chicken: Set out three large shallow bowls. Place the flour in the first bowl. Beat the eggs and mustard in the second bowl. In the third bowl, mix the panko, butter, Parmigiano Reggiano cheese, and the remaining ¼ teaspoon each of salt and pepper. Set up a breading station in this order: flour, egg/mustard mixture, panko mixture.
7. Remove the plastic wrap from the chicken cordon bleu rolls. Gently dredge each roll in the flour, coating evenly all around; dip in the egg/mustard mixture; then dredge in the panko mixture, turning and patting to adhere. Place the breaded chicken on the prepared baking sheet. Bake for about 25 minutes, until lightly golden and cooked through. Allow the chicken to rest on the baking sheet for 5 minutes before serving with the sauce. If you wish to slice it into pinwheels, use a serrated knife with a gentle sawing motion, as the breading can be delicate.

For the Sauce

1. While the chicken bakes, make the sauce. In a medium skillet over medium heat, melt the butter. Add the shallots and cook, stirring occasionally, until softened, 2 to 3 minutes. Stir in the flour and cook, stirring constantly, for about 1 minute. Add the chicken stock, stirring to combine, and then bring to a boil. Add the heavy cream, salt, pepper, mustard, and Parmigiano Reggiano and simmer until thickened, about 2 minutes. Add the thyme, then taste and adjust seasoning if necessary.
2. Make-Ahead Instructions: Prepare the chicken by rolling it with the filling up to a day in advance. If adding breading, do so no more than 3 hours before cooking. Once breaded, store the chicken uncovered in the refrigerator and then bake as directed.