Preheat oven to 375 degrees. Line baking sheet with parchment paper or grease.
Roll frozen dough into a rectangle, about 9" x 12". Place on baking sheet. Cut 1" strips from the edges of the dough to about 1/3 of the way to the center on both sides. Cut strips all the way up the long side of the dough.
Place deli ham down the center of the bread.
Sprinkle chicken over ham.
Sprinkle with Swiss cheese.
Cross strips over filling to create a braid.
Whisk egg and brush it on the braid. Crush dry stuffing mix until it's crumbly and sprinkle over egg wash.
Bake for 18-20 minutes.
While braid is baking make this super easy sauce for serving the Chicken Cordon Bleu Braid with.
In a saucepan warm up sour cream, cream of chicken soup, parmesan, mustard, and thyme. Add enough milk to make it a good consistency.
Cut braid in slices and serve with sauce!