In a saucepan, bring the broth and mushrooms (if using) to a boil. Remove from heat. Meanwhile, in a large skillet, over medium-low heat, heat the oil or butter. Add the onions, salt, and pepper. Cook for 8 minutes. Add the celery and cook for 7 minutes. Add the parsley and dill.
Heat oven to 375° F. Place the bread in a large bowl. Add the onion mixture and the broth and mushrooms (if using) and combine. Transfer to a 9-inch square or 11-by-7-inch baking dish or the cavity of the turkey (see Tip below).
Bake the stuffing for 25 minutes. If you prefer a soft stuffing, cover the dish with foil while baking. If you prefer a crusty stuffing, bake uncovered. (You can cover and refrigerate the uncooked stuffing for up to 24 hours. Bake as directed, adding 10 minutes to the baking time.)