Add all sausage to a large nonstick skillet and cook sausage 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 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 will face down so the stuffing will not stick) and bake 30 minutes. Uncover and bake 30 more minutes.