Place potatoes in a large pot and pour in water to cover by 2"; season generously with salt. Bring to a boil over medium-high heat, then reduce heat and simmer gently until potatoes are tender on the outside but still very firm in the center, 8–10 minutes.

Drain potatoes; transfer to a rimmed baking sheet and chill until cool, at least 1 hour.

Grate potatoes on the large holes of a box grater or with a food processor fitted with the grater attachment. Transfer grated potatoes to a large bowl and toss with 2 oz. grated Parmesan, 1½ tsp. Diamond Crystal or 1 tsp. Morton kosher salt, and 2 tsp. pepper.

Heat 3 Tbsp. oil in a medium nonstick skillet over medium. Add potato mixture and spread out in an even layer, pressing down to ensure potatoes make good contact with pan. Cook, occasionally pressing down on potatoes and running spatula around edges and bottom of pan to prevent sticking, until underside is evenly golden brown and crisp, 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 plate. Set pan back over medium heat, add remaining 1 Tbsp. oil, and carefully slide in rösti, crispy side up. 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 rösti freely move around in pan. Carefully slide rösti onto a platter and top with more Parmesan and pepper. Slice into wedges to serve.

Do Ahead: Potatoes can be boiled 1 day ahead. Let cool; cover and chill. Rösti (without additional cheese) can be made 3 hours ahead; store uncovered at room temperature. Reheat in a 300° oven 10 minutes before serving.