Preheat oven to 350ºF. Grease a 13-by-9-inch casserole dish; lightly coat one side of a large piece of foil with butter. In a large skillet over medium-high heat, warm oil. Add sausage and cook, stirring and breaking up into small pieces with a wooden spoon, until no longer pink, about 8 minutes. Use a slotted spoon to remove sausage to a large bowl; reserve fat in skillet.
Add onion and celery to skillet and cook 5 minutes, stirring. Add garlic, apples, sage and parsley and cook 5 minutes longer, stirring often. Remove from heat and add mixture to sausage.
Add bread cubes to sausage mixture, toss until well combined. Season with salt and pepper and toss again. Gently stir in chicken broth, then egg. Spoon mixture into casserole dish, cover with buttered foil and bake for 40 minutes. Remove foil and bake about 20 minutes longer, until stuffing is crisp and golden on top. Serve warm.