Preheat the oven to 375 degrees F.
On a clean dry work surface sprinkled with flour, roll the dough into a 10 by 14-inch rectangle, making sure not to flip the dough over, just rotating by 90 degrees as needed. Once the dough is rolled out, sprinkle the mushrooms in the center, leaving a 2-inch border on all sides. Top with the beef
, followed by half the cheese, then the chicken, in a stacked fashion. Top with the herbs
, tomato sauce
, and remaining cheese
. Brush the edges with some of the egg wash. Fold the bottom and top edges over the filling, then fold the sides over twice, 1-inch at a time, resulting in a 6 by 14-inch tube. Brush the outside of the stromboli
with the remaining egg wash
, place on a sheet pan
lined with parchment paper, and bake until firm and golden brown, 15 to 18 minutes. Remove from the oven and let rest for 5 minutes. Cut into 5 to 6 portions and serve.