Heat oven to 350° F. Butter a 9-by-13-inch baking dish. Tear the buns into large pieces and place in the dish; set aside.
Heat the oil in a large skillet over medium-high heat. Add the onion and cook for 1 minute. Add the sausage meat to the skillet. Cook, breaking up the sausage with a wooden spoon, until no trace of pink remains, about 10 minutes.
In a large bowl, whisk together the eggs, milk, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Stir in the cooked sausage and onion along with the parsley and 1 cup of the cheese. Pour the mixture over the buns, pressing with a spatula or spoon to submerge them. Sprinkle with the remaining cheese. Bake until the eggs are set, about 35 minutes. Let cool for 5 minutes before serving.