Preheat oven to 375°. Place potatoes in a large saucepan and cover with water. Bring to a boil and reduce heat and simmer until almost fork tender, about 6 to 7 minutes. Drain.

While potatoes are cooking, in a large skillet, cook and crumble sausage with onion over medium heat until no longer pink, 6-8 minutes. Remove from pan with a slotted spoon and place on a plate lined with a paper towel. If there is a lot of grease, drain off all but about a tablespoon.

In that same skillet, heat butter over medium-high heat. Add potatoes and sprinkle with salt and pepper. Cook until golden brown, 10-15 minutes, stirring occasionally. Add cook sausage and mix until just combined.

With the back of a spoon, make seven or eight wells in the potato mixture. Break one egg into each well. Sprinkle with cheese.

Bake until egg whites are set and yolks begin to thicken but are not hard, 12-15 minutes. Sprinkle with parsley and season with more salt and pepper if needed.