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.