Rösti is a large potato pancake made famous by the Swiss. This version omits the speck, but feel free to add a slice under the cheese. Chilling parboiled potatoes makes the grating step easier.
Place potatoes in a large saucepan, add cold water to cover, and season generously with salt. Bring to a boil; reduce heat and simmer until the tip of a paring knife slides easily about 1/2" into potatoes, 8-10 minutes. Drain potatoes and let cool. Chill until firm, about 2 hours.
Preheat oven to 300°F. Peel potatoes. Using the large holes of a box grater, grate potatoes. Melt 4 tablespoons butter in a 12" ovenproof skillet over medium-low heat. Add potatoes. Season with salt and pepper; press gently to compact. Cook, occasionally shaking pan to loosen, until bottom is golden brown and crisp, 15-20 minutes.
Slide rösti onto a plate. Carefully invert skillet over plate and flip to return rösti to pan, browned side up. Dot 1 tablespoon butter around edge, allowing it to melt around and under rösti. Season with salt and pepper and cook until second side is golden brown and center is tender, 10-15 minutes longer.
Top rösti with cheese and bake until cheese melts, 5-8 minutes.
Meanwhile, melt remaining 1 tablespoon butter in a large nonstick skillet over medium-low heat. Crack eggs into skillet and cook until whites are just set, about 4 minutes.
Cut rösti into quarters, top with eggs, and garnish with parsley.
