Place potatoes in a large pot and pour in water to cover by 2 inches. Season generously with salt. Bring to a boil over medium-high heat and then reduce heat and simmer gently until potatoes or tender on the outside but still very firm in the center, 8-10 minutes
Drain potatoes and transfer to a rimmed baking sheet. Chill until cool, at least 1 hour
Grate potatoes on large hole of a box grater or with a food processor fitted with the grater attachment. Transfer grated potatoes to a large bowl and toss with Parmesan, salt, and pepper
Heat 3 tbsp oil in a medium nonstick skillet over medium. Add potato mixture and, using a heatproof spatula, spread out into an even layer, pressing down to ensure potatoes make good contact with the pan. Cook, occasionally pressing down on the potatoes and running spatula around edges and bottom of pan to prevent sticking, until underside is evenly golden brown and crispy, about 7-9 minutes. Remove from heat
Set a large plate upside down over pan. In one swift motion, flip pan and plate to invert rösti onto the plate. Set pan back over medium heat, add remaining 1 tbsp oil to pan, and carefully slide rösti, crispy side up, back into the pan. Using spatula, tuck edges into pan to create a neat disk. Cook until second side is golden brown, 5-7 minutes. Run spatula around edges of rösti and give pan a shake until you can feel the rösti moving freely in the pan. Carefully slide rösti onto a platter and top with more Parmesan and black pepper
Slice into wedges and serve