Preheat the oven to 375°F.
Add 1 tablespoon of the butter to a large ovenproof braiser, cast iron skillet, or sauté pan over medium heat. (If you don’t have an ovenproof pan, don’t panic—you can transfer your sausage to a casserole dish later on.) When the butter is melted and bubbling, add the sausage to the pan and cook for about 6 minutes, stirring and breaking up the meat into small pieces with a spatula, until lightly browned.
Stir in 3 more tablespoons of the butter. Once melted, stir in the apple, onion, and thyme (if using). Pour in ½ cup of the chicken broth to deglaze, and use your spatula to scrape up all the delicious browned bits from the bottom of the pan. Reduce the heat to medium-low and cook for 6 to 8 minutes until the onion is translucent and the apple is soft.
Add the remaining 2½ cups of broth to the pan and bring it to a boil. Turn off the heat and fold in the Stovetop stuffing mix. Cover the pan with a lid and let it sit for 5 minutes, then fluff the stuffing with a fork. (If you’re transferring the stuffing to a baking dish, do that now.)
Dot the remaining butter on top of the stuffing and bake for 15-20 minutes or until browned and crisped on top to your liking. Serve warm topped with fresh parsley if you feel fancy.