Rosti with Fried Eggs

3 russet potatoes (about 1 ½ lb)
Kosher salt
6 Tbsp. (3/4 stick) unsalted butter, divided
freshly ground black pepper
4 oz raclette or gruyere cheese, sliced
4 large eggs
chopped flat-leaf parsley

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 ½” into potatoes, 8-10 minutes. Drain potatoes and let cool. Chill until firm, about 2 hours.

Preheat oven to 300 degrees. Peel potatoes. Using the large holes of a box grater, grate potatoes. Melt 4 Tbsp butter in a 12″ oven proof 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 Rosti onto a plate. Carefully invert skillet over plate and flip to return Rosti to pan, browned side up. Dot 1 Tbsp butter around edges, allowing it to melt around and under Rosti. Season with salt and pepper and cook until second side is golden brown and center is tender, 10-15 minutes

Top Rosti with cheese and bake until cheese melts, 5-8 minutes.

Meanwhile, melt remaining 1 Tbsp butter in a large non stick skillet over medium-low heat. Crack eggs into skillet and cook until whites are just set, about 4 minutes.

Cut Rosti into quarters, top with eggs, garnish with parsley.