1. Preheat oven to 375 degrees F (190 degrees C).
2. Mix mayonnaise, Dijon mustard, and black pepper together in a bowl.
3. Lightly toss together cooked chicken, celery, parsley, cranberries, and Swiss cheese in a separate bowl. Stir the mayonnaise mixture into the turkey (or chicken) mixture until well combined.
4. Separate crescent roll dough into 16 triangles.
5. Place 8 dough triangles onto a large baking pan, arranging them in a circle with the wide ends together in the center. Pinch the corners of the wide ends together to join the circle.
6. Place a new dough triangle down next to an existing one with the pointed end towards the center. Pinch the two triangles together to make a rectangular-shaped petal. Repeat with remaining 7 triangles to make 8 rectangular dough petals in a circle.
7. Scoop up about 1/2 cup of turkey (or chicken) mixture and place at the bottom of a dough rectangle towards the center.
8. Brush the wreath with beaten egg.
9. Bake in the preheated oven until the filling is hot and the crescent roll dough is golden brown, 25 to 30 minutes.