In a 12-in. frying pan over high heat, cook mushrooms, butter, onions, celery, and garlic, stirring often, until vegetables are lightly browned, about 15 minutes. Pour into a large bowl. Add a bit of broth to pan and stir to scrape up browned bits. Add to bowl.
Pour 2 cups broth into bowl and add bread, artichoke hearts, parmesan, poultry seasoning, and rosemary; mix well. Add salt and pepper to taste. Make a well in stuffing. Add egg and beat with a fork to blend; mix egg with stuffing.
Preheat oven to 325° to 350° (use temperature turkey requires; see Note below). Spoon stuffing into a shallow 3-qt. (9- by 13-in.) casserole. For moist stuffing, cover with foil; for crusty stuffing, do not cover. Bake until hot (at least 150° in center) or lightly browned, about 50 minutes.
Make ahead: Up to 1 day ahead, make stuffing, put in casserole, cover, and chill. Allow about 1 hour to bake.
Note: For turkeys 10-13 lbs., oven/bbq temperature should be 350°; for turkeys 14 lbs. and over, oven/bbq temperature should be 325°.