Melt butter over medium-high heat in a skillet, add the sausage, break into pieces, and cook until golden brown, about 7 minutes. Reduce the heat to medium, add the onion, garlic, celery, and a pinch of salt, and sauté, stirring frequently, for 5 minutes. Add the mushrooms and a pinch of salt and cook for 3 minutes.
Place the bread in a large bowl and stir in the sausage mixture. Add 3 cups broth and marjoram and mix well (if the mixture is still dry, add a little more broth). Season to taste with salt and pepper. Butter a 9-by-11-inch baking dish and transfer the stuffing to the dish, cover with foil, and place in the oven for 15 minutes. Remove the foil and cook for about 45 minutes, until the top is crispy and brown.