Dice the onion. Whisk the eggs in a bowl and season with salt. Slice the potatoes into sticks or roughly grate them, then fold in the diced onion.
Heat the rapeseed oil in a pan over medium-high heat. Add the diced ham and fry for 1 minute. Add the potato mixture, fry for 1-2 minutes while turning, season with pepper, and shape into a hash brown patty. Spread the grated cheese over the hash brown, then pour the whisked eggs over it. Cover the pan and cook over low heat for 5-6 minutes until the underside is browned and the eggs have set. Using a flat lid or plate, carefully turn the hash brown and fry for another 4-5 minutes until the second side is golden brown.
Cut the chives into thin slices and sprinkle over the rösti before serving.