1. Heat oven to 350°F. Butter a 9-inch-square glass baking dish.
2. Place potatoes in a large saucepan with cold water to cover. Bring to boiling over high heat. Reduce heat and simmer until tender, about 10 minutes. Drain and transfer potatoes to prepared pan. Sprinkle sausage and cheese over potatoes.
3. In a medium bowl, whisk together eggs, scallions, salt, and pepper and pour into pan.
4. Bake until golden and just set in center, 35 to 40 minutes. Let stand 5 minutes before serving.