Spread bread cubes in a single layer in 2 pans, each 10 by 15 inches. Bake in a 350° oven until toasted golden brown, about 25 minutes. Turn cubes over with a wide spatula occasionally. After 15 minutes, switch pan positions.
Trim and discard discolored mushroom stem ends. Slice mushrooms.
In a 10- to 12-inch frying pan over high heat, combine butter, mushrooms, onions, celery, and garlic. Stir often until vegetables are lightly browned, about 15 minutes. Pour into a large bowl. Add a little broth to pan and stir to scrape browned bits free. Add to bowl.
Pour 2 cups broth into bowl and add toasted bread, artichoke hearts, cheese, poultry seasoning, and rosemary; mix well. Add salt and pepper to taste. Make a well in dressing, add egg, beat with a fork to blend, then mix egg with dressing.
Spoon into a shallow 3-quart (9- by 13-in.) casserole. For moist dressing, cover with foil; for crusty dressing, do not cover. Bake in a 325° to 350° oven until hot (at least 150° in center) or lightly browned, about 50 minutes (1 hour if chilled).