Rösti with Fried Eggs

Chris saw this recipe in the February Bon Appetit magazine and he fixed it for our dinner one night. It was very good and we decided to add it to the blog so that we wouldn't forget to make it again. It is great for a simple dinner for 4.

Rösti with Fried Eggs  
3 russet potatoes (about 1.5 lbs)
kosher salt
6 Tbsp. unslated butter, divided
freshly ground black pepper
4 oz. raclette or Gruyère cheese, sliced (we used Gruyère)
4 large eggs
chopped flat-leaf parsley
Place potatoes in a large saucepan and add cold water to cover. 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 the potatoes, 8-10 mins. Drain potatoes and let cool. Chill until firm, about 2 hrs.
Preheat oven to 300º. Peel potatoes and grate using the large holes of a box grater. Melt 4 Tbsp. butter in a 12 ovenproof skillet over med-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 mis.
Slide rösti onto a plate. Carefully invert skillet over plate and flip to return it to the pan, browned side up. Dot 1 Tbsp. butter around edge, allowing it to melt around and under rösti. Season with salt and pepper and cook until second side is golden and center is tender, 10-15 mins. longer.
Top rösti with cheese and bake until cheese melts, 5-8 mins.
Melt remaining 1 Tbsp butter in a large nonstick skillet over medium-low heat and add eggs, cooking until whites are just set, about 4 mins.
Cut rösti into quarter, top with eggs, and garnish with parsley.