Add all sausage to a large nonstick skillet and cook over medium-high heat until browned and cooked through, breaking up bulk sausage with a wooden spoon. Using a slotted spoon, remove sausage to a plate and set aside.
Add 1 tablespoon butter to the pan with sausage grease and add onions, celery, and red peppers. Cook until softened, about 5 minutes.
Stir in green onions, hot sauce, thyme, and sage. Season with salt and pepper. Add sausage back to skillet and stir to combine.
Transfer sausage mixture to a large bowl and stir in stuffing mix. Season with salt and pepper.
Preheat oven to 350 degrees. Butter a 13×9-inch baking dish. Add enough broth to stuffing mix to moisten well (3/4 to 1 1/2 cups). Transfer to baking dish and scatter butter pieces across top.
Cover with foil (butter the side that faces down) and bake for 30 minutes. Uncover and bake 30 more minutes.
