Preheat the oven to 350°F. Grease a 13-by-9-inch (or other 3-quart) baking dish with butter or nonstick cooking spray.

Heat a large skillet over medium-high heat. Add the sausage, breaking it up into small pieces with a wooden spoon. Cook the meat until no pink pieces remain, 6 to 8 minutes. Add the bell pepper and white and light green parts of the green onion, and cook for 3 more minutes. Transfer the mixture to a plate to cool slightly.

Meanwhile, in a large bowl, whisk together the eggs, heavy cream, mustard, hot sauce, if using, salt, black pepper, and about one-half of the remaining green parts of the green onion. Gently fold 1 ½ cups of cheese, the hash browns, and the sausage mixture into the egg mixture. Transfer the mixture to the baking dish and smooth it into an even layer.

Cover with aluminum foil and bake for 50 minutes. Remove the foil and top with the remaining ½ cup of cheese. Increase the oven temperature to 400°F and finish baking until the center of the casserole is set (not jiggly) and the top is slightly golden, 10 to 15 minutes more. Let rest for 10 minutes, top with the remaining green parts of the green onion, and serve.