Step 1

Preheat oven to 400°. Grate potatoes on the large holes of a box grater. Transfer to a fine-mesh sieve and add onion; squeeze to expel as much water from potatoes as possible. If you want, wrap in a kitchen towel to help soak up the excess water. Sprinkle with 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, season with pepper; toss to combine.

Step 2

Heat ¼ cup oil in a large ovenproof nonstick skillet over medium-high. Add potato mixture and press firmly into an even layer. Cook, shaking pan a few times (potatoes may stick at first, but just keep going; they will release), until deeply browned underneath (carefully lift with a spatula to peek), 8–10 minutes. Transfer to oven; bake until rosti is cooked through (pluck a bit off the top and taste for doneness), 18–20 minutes.

Step 3

Meanwhile, whisk garlic, lemon juice, mayonnaise, mustard, and remaining ¼ cup oil in a large bowl until smooth. Whisk in 2 Tbsp. Parmesan and season with salt and pepper. Set dressing aside.

Step 4

When rosti is cooked through, remove from oven and evenly crack eggs on top; season with salt and pepper. Bake until eggs are just set, 10–12 minutes.

Step 5

Add arugula to reserved dressing and toss to coat. Top rosti with salad.